TPTP Problem File: ITP278^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP278^2 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Uniqueness 00535_035038
% 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 : 0075_VEBT_Uniqueness_00535_035038 [Des22]
% Status : Theorem
% Rating : 0.67 v8.1.0
% Syntax : Number of formulae : 9773 (3324 unt; 806 typ; 0 def)
% Number of atoms : 28895 (9740 equ; 4 cnn)
% Maximal formula atoms : 71 ( 3 avg)
% Number of connectives : 208649 (2570 ~; 321 |;2320 &;190319 @)
% ( 0 <=>;13119 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 8 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 5249 (5249 >; 0 *; 0 +; 0 <<)
% Number of symbols : 798 ( 794 usr; 28 con; 0-9 aty)
% Number of variables : 32530 (3050 ^;27787 !; 930 ?;32530 :)
% ( 763 !>; 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
% 2022-02-18 16:35:29.167
%------------------------------------------------------------------------------
% 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 (787)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Odvd,type,
dvd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Oring__gcd,type,
ring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_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_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_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_Parity_Oring__parity,type,
ring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__lattice,type,
finite_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Osemiring__numeral,type,
semiring_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__modulo,type,
semidom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__modulo,type,
semiring_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__parity,type,
semiring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring__abs,type,
ordered_ring_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oalgebraic__semidom,type,
algebraic_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__cancel,type,
semiring_1_cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Odistrib__lattice,type,
distrib_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Obanach,type,
real_Vector_banach:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring,type,
linordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Partial__Order_Occpo,type,
comple9053668089753744459l_ccpo:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__distrib__lattice,type,
finite8700451911770168679attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot1__space,type,
topological_t1_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot2__space,type,
topological_t2_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot3__space,type,
topological_t3_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot4__space,type,
topological_t4_space:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__group__add,type,
topolo1633459387980952147up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
comm_s4317794764714335236cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord4710134922213307826strict:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1802427076303600483id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__monoid__add,type,
topolo6943815403480290642id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__field,type,
real_V7773925162809079976_field:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_15535105094025558882visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel2418104881723323429up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__monoid__mult,type,
topolo1898628316856586783d_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
real_V4867850818363320053vector:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim2362893244070406136eiling:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd__mult__normalize,type,
semiri6843258321239162965malize:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere166539214618696060dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere6658533253407199908up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__ab__group__add,type,
topolo1287966508704411220up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V7819770556892013058_space:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_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_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_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_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_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
euclid8789492081693882211th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
euclid5411537665997757685th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Countable__Complete__Lattices_Ocountable__complete__lattice,type,
counta3822494911875563373attice:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
thf(sy_cl_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_OCsum,type,
bNF_Cardinal_Csum:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
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_Ocfinite,type,
bNF_Cardinal_cfinite:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
bNF_Ca4139267488887388095finite:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
bNF_Cardinal_cone: set @ ( product_prod @ product_unit @ product_unit ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocprod,type,
bNF_Cardinal_cprod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_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_Octwo,type,
bNF_Cardinal_ctwo: set @ ( product_prod @ $o @ $o ) ).
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_OisCardSuc,type,
bNF_Ca6246979054910435723ardSuc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ 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_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_Ocollect,type,
bNF_collect:
!>[B: $tType,A: $tType] : ( ( set @ ( B > ( set @ A ) ) ) > B > ( set @ A ) ) ).
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_OfromCard,type,
bNF_Gr5436034075474128252omCard:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > B > A ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_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,type,
bNF_Greatest_toCard:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > A > B ) ).
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_Odir__image,type,
bNF_We2720479622203943262_image:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > A2 ) > ( set @ ( product_prod @ A2 @ A2 ) ) ) ).
thf(sy_c_BNF__Wellorder__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__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_Oand__int__rel,type,
bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Bit__Operations_Oand__not__num,type,
bit_and_not_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
bit_and_not_num_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: num > num > num ).
thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
bit_or3848514188828904588eg_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > nat > $o ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
bit_un4731106466462545111um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
bit_un2480387367778600638or_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
bit_un2901131394128224187um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
boolea2506097494486148201lgebra:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
thf(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
thf(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
code_integer_of_nat: nat > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__num,type,
code_integer_of_num: num > code_integer ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onegative,type,
code_negative: num > code_integer ).
thf(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
thf(sy_c_Code__Numeral_Opositive,type,
code_positive: num > code_integer ).
thf(sy_c_Code__Target__Int_Onegative,type,
code_Target_negative: num > int ).
thf(sy_c_Code__Target__Int_Opositive,type,
code_Target_positive: num > int ).
thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_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_Complex_Orcis,type,
rcis: real > real > complex ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
condit941137186595557371_above:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
condit1013018076250108175_below:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_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_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_Deriv_Ohas__vector__derivative,type,
has_ve8173657378732805170vative:
!>[B: $tType] : ( ( real > B ) > B > ( filter @ real ) > $o ) ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: ( product_prod @ int @ int ) > int ).
thf(sy_c_Divides_Oadjust__mod,type,
adjust_mod: int > int > int ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
thf(sy_c_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_Extended__Nat_OeSuc,type,
extended_eSuc: extended_enat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat,type,
extended_enat2: nat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
extended_case_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
extend4730790105801354508finity:
!>[A: $tType] : A ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_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_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Finite__Set_Ofold__graph,type,
finite_fold_graph:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B > $o ) ).
thf(sy_c_Finite__Set_Ofolding__idem__on,type,
finite1890593828518410140dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__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_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Fun__Def_Omax__strict,type,
fun_max_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omax__weak,type,
fun_max_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omin__strict,type,
fun_min_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omin__weak,type,
fun_min_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Opair__leq,type,
fun_pair_leq: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Fun__Def_Opair__less,type,
fun_pair_less: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Fun__Def_Oreduction__pair,type,
fun_reduction_pair:
!>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
thf(sy_c_Fun__Def_Orp__inv__image,type,
fun_rp_inv_image:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
thf(sy_c_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > ( product_prod @ int @ int ) ).
thf(sy_c_GCD_Obezw__rel,type,
bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Osemiring__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_Groups__List_Omonoid__mult__class_Oprod__list,type,
groups5270119922927024881d_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_HOL_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_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_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_Oat__infinity,type,
at_infinity:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_List_OBleast,type,
bleast:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).
thf(sy_c_List_Oabort__Bleast,type,
abort_Bleast:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
thf(sy_c_List_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_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Ocoset,type,
coset:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
thf(sy_c_List_Oextract,type,
extract:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
thf(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
thf(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ).
thf(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( B > A > B ) > B > ( list @ A ) > B ) ).
thf(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_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_Olexn,type,
lexn:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olinorder_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
sorted8670434370408473282of_set:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__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_Osort__key,type,
linorder_sort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord4507533701916653071of_set:
!>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Olist__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_Olistrel1p,type,
listrel1p:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olists,type,
lists:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__project,type,
map_project:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_List_Omin__list,type,
min_list:
!>[A: $tType] : ( ( list @ 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_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OremoveAll,type,
removeAll:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_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_Osorted__wrt__rel,type,
sorted_wrt_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_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__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__comp,type,
map_comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > ( option @ C ) ) > ( A > ( option @ B ) ) > A > ( option @ C ) ) ).
thf(sy_c_Map_Omap__of,type,
map_of:
!>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Omap__upds,type,
map_upds:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Oran,type,
ran:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Map_Orestrict__map,type,
restrict_map:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > A > ( option @ B ) ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Nat_Ofunpow,type,
funpow:
!>[A: $tType] : ( nat > ( A > A ) > A > A ) ).
thf(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
thf(sy_c_Nat_Onat_Opred,type,
pred: nat > nat ).
thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
rec_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
thf(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
rec_set_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T > $o ) ).
thf(sy_c_Nat_Osemiring__1__class_ONats,type,
semiring_1_Nats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_Olist__decode,type,
nat_list_decode: nat > ( list @ nat ) ).
thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
nat_list_decode_rel: nat > nat > $o ).
thf(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: ( list @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
nat_list_encode_rel: ( list @ nat ) > ( list @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__decode,type,
nat_prod_decode: nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: ( product_prod @ nat @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > ( set @ nat ) ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: ( set @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_NthRoot_Osqrt,type,
sqrt: real > real ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Osub,type,
neg_numeral_sub:
!>[A: $tType] : ( num > num > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum_Ocase__num,type,
case_num:
!>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_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_Omap__option,type,
map_option:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( option @ A ) > ( option @ Aa ) ) ).
thf(sy_c_Option_Ooption_Orec__option,type,
rec_option:
!>[C: $tType,A: $tType] : ( C > ( A > C ) > ( option @ A ) > C ) ).
thf(sy_c_Option_Ooption_Osize__option,type,
size_option:
!>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
thf(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Option_Othese,type,
these:
!>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Order__Continuity_Ocountable__complete__lattice__class_Occlfp,type,
order_532582986084564980_cclfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Order__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_OAbove,type,
order_Above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OAboveS,type,
order_AboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OUnder,type,
order_Under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OUnderS,type,
order_UnderS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OaboveS,type,
order_aboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Olinear__order__on,type,
order_679001287576687338der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_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_Orelation__of,type,
order_relation_of:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
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__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
product_rec_set_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
thf(sy_c_Product__Type_Ounit_OAbs__unit,type,
product_Abs_unit: $o > product_unit ).
thf(sy_c_Product__Type_Ounit_ORep__unit,type,
product_Rep_unit: product_unit > $o ).
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_OFrct,type,
frct: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_ORep__Rat,type,
rep_Rat: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
field_char_0_of_rat:
!>[A: $tType] : ( rat > A ) ).
thf(sy_c_Rat_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oof__int,type,
of_int: int > rat ).
thf(sy_c_Rat_Opcr__rat,type,
pcr_rat: ( product_prod @ int @ int ) > rat > $o ).
thf(sy_c_Rat_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,
positive: 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_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_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_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_Oconversep,type,
conversep:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > A > $o ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_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_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_Oascii__of,type,
ascii_of: char > char ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : ( char > A ) ).
thf(sy_c_String_Ointeger__of__char,type,
integer_of_char: char > code_integer ).
thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : ( A > char ) ).
thf(sy_c_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_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_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_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_Oacyclic,type,
transitive_acyclic:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Transitive__Closure_Ontrancl,type,
transitive_ntrancl:
!>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_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_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
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_Orec__VEBT,type,
vEBT_rec_VEBT:
!>[A: $tType] : ( ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ ( product_prod @ vEBT_VEBT @ A ) ) > vEBT_VEBT > A > A ) > ( $o > $o > A ) > vEBT_VEBT > A ) ).
thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
vEBT_V312737461966249ad_rel: ( product_prod @ vEBT_VEBT @ extended_enat ) > ( product_prod @ vEBT_VEBT @ extended_enat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
vEBT_V5917875025757280293ildren: nat > ( list @ vEBT_VEBT ) > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
vEBT_V4351362008482014158ma_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
vEBT_V5765760719290551771er_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
vEBT_VEBT_valid_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_Oset__vebt,type,
vEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(sy_c_VEBT__Delete_Ovebt__delete,type,
vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
vEBT_vebt_delete_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H,type,
vEBT_VEBT_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H__rel,type,
vEBT_VEBT_insert_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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_VEBT__MinMax_OVEBT__internal_Oadd,type,
vEBT_VEBT_add: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
vEBT_VEBT_greater: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
vEBT_VEBT_less: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
vEBT_VEBT_lesseq: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
vEBT_VEBT_max_in_set: ( set @ nat ) > nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
vEBT_VEBT_min_in_set: ( set @ nat ) > nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
vEBT_VEBT_mul: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift,type,
vEBT_V2048590022279873568_shift:
!>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
vEBT_VEBT_power: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
vEBT_vebt_maxt: vEBT_VEBT > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
vEBT_vebt_mint: vEBT_VEBT > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
vEBT_is_pred_in_set: ( set @ nat ) > nat > nat > $o ).
thf(sy_c_VEBT__Pred_Ovebt__pred,type,
vEBT_vebt_pred: vEBT_VEBT > nat > ( option @ nat ) ).
thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
vEBT_vebt_pred_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
vEBT_is_succ_in_set: ( set @ nat ) > nat > nat > $o ).
thf(sy_c_VEBT__Succ_Ovebt__succ,type,
vEBT_vebt_succ: vEBT_VEBT > nat > ( option @ nat ) ).
thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
vEBT_vebt_succ_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
thf(sy_c_Wellfounded_Oless__than,type,
less_than: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__extp,type,
max_extp:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Zorn_OChains,type,
chains:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
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_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_deg____,type,
deg: nat ).
thf(sy_v_info____,type,
info: option @ ( product_prod @ nat @ nat ) ).
thf(sy_v_m____,type,
m: nat ).
thf(sy_v_ma____,type,
ma: nat ).
thf(sy_v_mi____,type,
mi: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_sa____,type,
sa: vEBT_VEBT ).
thf(sy_v_summary_H____,type,
summary: vEBT_VEBT ).
thf(sy_v_summary____,type,
summary2: vEBT_VEBT ).
thf(sy_v_treeList_H____,type,
treeList: list @ vEBT_VEBT ).
thf(sy_v_treeList____,type,
treeList2: list @ vEBT_VEBT ).
% Relevant facts (8183)
thf(fact_0__092_060open_062summary_A_061_Asummary_H_092_060close_062,axiom,
summary2 = summary ).
% \<open>summary = summary'\<close>
thf(fact_1_case4_I9_J,axiom,
ord_less_eq @ nat @ mi @ ma ).
% case4(9)
thf(fact_2_infsplit,axiom,
( info
= ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) ) ).
% infsplit
thf(fact_3_case4_I13_J,axiom,
( ( vEBT_VEBT_set_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ deg @ treeList2 @ summary2 ) )
= ( vEBT_VEBT_set_vebt @ sa ) ) ).
% case4(13)
thf(fact_4_case4_I12_J,axiom,
vEBT_invar_vebt @ sa @ deg ).
% case4(12)
thf(fact_5_VEBT_Oinject_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,Y11: option @ ( product_prod @ nat @ nat ),Y12: nat,Y13: list @ vEBT_VEBT,Y14: vEBT_VEBT] :
( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
= ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 )
& ( X13 = Y13 )
& ( X14 = Y14 ) ) ) ).
% VEBT.inject(1)
thf(fact_6_option_Oinject,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( ( some @ A @ X2 )
= ( some @ A @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_7__092_060open_062treeList_A_061_AtreeList_H_092_060close_062,axiom,
treeList2 = treeList ).
% \<open>treeList = treeList'\<close>
thf(fact_8_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_9_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A3 @ B2 )
= ( product_Pair @ A @ B @ A4 @ B3 ) )
= ( ( A3 = A4 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_10_case4_I8_J,axiom,
( ( mi = ma )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ treeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) ) ) ).
% case4(8)
thf(fact_11__092_060open_062set__vebt_H_Asummary_A_061_Aset__vebt_H_Asummary_H_092_060close_062,axiom,
( ( vEBT_VEBT_set_vebt @ summary2 )
= ( vEBT_VEBT_set_vebt @ summary ) ) ).
% \<open>set_vebt' summary = set_vebt' summary'\<close>
thf(fact_12__092_060open_062_092_060And_062x_O_Avebt__member_Asummary_Ax_A_061_Avebt__member_Asummary_H_Ax_092_060close_062,axiom,
! [X3: nat] :
( ( vEBT_vebt_member @ summary2 @ X3 )
= ( vEBT_vebt_member @ summary @ X3 ) ) ).
% \<open>\<And>x. vebt_member summary x = vebt_member summary' x\<close>
thf(fact_13_aa,axiom,
ord_less_eq @ ( set @ nat ) @ ( insert2 @ nat @ mi @ ( insert2 @ nat @ ma @ ( bot_bot @ ( set @ nat ) ) ) ) @ ( vEBT_VEBT_set_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ deg @ treeList2 @ summary2 ) ) ).
% aa
thf(fact_14_prod__decode__aux_Ocases,axiom,
! [X3: product_prod @ nat @ nat] :
~ ! [K: nat,M: nat] :
( X3
!= ( product_Pair @ nat @ nat @ K @ M ) ) ).
% prod_decode_aux.cases
thf(fact_15_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_16_deg__deg__n,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( Deg = N ) ) ).
% deg_deg_n
thf(fact_17_both__member__options__equiv__member,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X3 )
= ( vEBT_vebt_member @ T2 @ X3 ) ) ) ).
% both_member_options_equiv_member
thf(fact_18_valid__member__both__member__options,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X3 )
=> ( vEBT_vebt_member @ T2 @ X3 ) ) ) ).
% valid_member_both_member_options
thf(fact_19_max__in__set__def,axiom,
( vEBT_VEBT_max_in_set
= ( ^ [Xs: set @ nat,X4: nat] :
( ( member @ nat @ X4 @ Xs )
& ! [Y3: nat] :
( ( member @ nat @ Y3 @ Xs )
=> ( ord_less_eq @ nat @ Y3 @ X4 ) ) ) ) ) ).
% max_in_set_def
thf(fact_20_min__in__set__def,axiom,
( vEBT_VEBT_min_in_set
= ( ^ [Xs: set @ nat,X4: nat] :
( ( member @ nat @ X4 @ Xs )
& ! [Y3: nat] :
( ( member @ nat @ Y3 @ Xs )
=> ( ord_less_eq @ nat @ X4 @ Y3 ) ) ) ) ) ).
% min_in_set_def
thf(fact_21_insert_H__pres__valid,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( vEBT_invar_vebt @ ( vEBT_VEBT_insert @ T2 @ X3 ) @ N ) ) ).
% insert'_pres_valid
thf(fact_22_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 )
=> ( ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) )
& ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_23_lesseq__shift,axiom,
( ( ord_less_eq @ nat )
= ( ^ [X4: nat,Y3: nat] : ( vEBT_VEBT_lesseq @ ( some @ nat @ X4 ) @ ( some @ nat @ Y3 ) ) ) ) ).
% lesseq_shift
thf(fact_24_case4_I3_J,axiom,
vEBT_invar_vebt @ summary2 @ m ).
% case4(3)
thf(fact_25_case4_I1_J,axiom,
! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ treeList2 ) )
=> ( ( vEBT_invar_vebt @ X @ na )
& ! [Xa: vEBT_VEBT] :
( ( vEBT_invar_vebt @ Xa @ na )
=> ( ( ( vEBT_VEBT_set_vebt @ X )
= ( vEBT_VEBT_set_vebt @ Xa ) )
=> ( Xa = X ) ) ) ) ) ).
% case4(1)
thf(fact_26_member__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_vebt_member @ T2 @ X3 )
= ( member @ nat @ X3 @ ( vEBT_set_vebt @ T2 ) ) ) ) ).
% member_correct
thf(fact_27_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_28_prod__induct7,axiom,
! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
=> ( P @ X3 ) ) ).
% prod_induct7
thf(fact_29_prod__induct6,axiom,
! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
=> ( P @ X3 ) ) ).
% prod_induct6
thf(fact_30_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X3 ) ) ).
% prod_induct5
thf(fact_31_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P @ X3 ) ) ).
% prod_induct4
thf(fact_32_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P @ X3 ) ) ).
% prod_induct3
thf(fact_33_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_34_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_35_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_36_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_37_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B4: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_38_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A3 @ B2 )
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ~ ( ( A3 = A4 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_39_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_40_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X5: A,Y4: B] :
( P2
= ( product_Pair @ A @ B @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_41_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A3: A,A6: set @ A] :
( ( ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ A @ A3 @ A6 ) )
= ( ( A3 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_42_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A6: set @ A,B2: A] :
( ( ( insert2 @ A @ A3 @ A6 )
= ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_43_buildup__gives__empty,axiom,
! [N: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% buildup_gives_empty
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
( ! [X5: A] :
( ( F3 @ X5 )
= ( G3 @ X5 ) )
=> ( F3 = G3 ) ) ).
% ext
thf(fact_48_insert__subset,axiom,
! [A: $tType,X3: A,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X3 @ A6 ) @ B5 )
= ( ( member @ A @ X3 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ A6 @ B5 ) ) ) ).
% insert_subset
thf(fact_49_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_50_subset__empty,axiom,
! [A: $tType,A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_51_empty__subsetI,axiom,
! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A6 ) ).
% empty_subsetI
thf(fact_52_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
= ( F1 @ A3 @ B2 ) ) ).
% old.prod.rec
thf(fact_53_case4_I2_J,axiom,
! [S: vEBT_VEBT] :
( ( vEBT_invar_vebt @ S @ m )
=> ( ( ( vEBT_VEBT_set_vebt @ summary2 )
= ( vEBT_VEBT_set_vebt @ S ) )
=> ( S = summary2 ) ) ) ).
% case4(2)
thf(fact_54_aca,axiom,
( ( size_size @ ( list @ vEBT_VEBT ) @ treeList )
= ( size_size @ ( list @ vEBT_VEBT ) @ treeList2 ) ) ).
% aca
thf(fact_55_subset__singletonD,axiom,
! [A: $tType,A6: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ( A6
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_56_subset__singleton__iff,axiom,
! [A: $tType,X6: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X6 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X6
= ( bot_bot @ ( set @ A ) ) )
| ( X6
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_57_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_58_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_59_all__not__in__conv,axiom,
! [A: $tType,A6: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A6 ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_60_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_61_subset__antisym,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( A6 = B5 ) ) ) ).
% subset_antisym
thf(fact_62_subsetI,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( member @ A @ X5 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B5 ) ) ).
% subsetI
thf(fact_63_insert__absorb2,axiom,
! [A: $tType,X3: A,A6: set @ A] :
( ( insert2 @ A @ X3 @ ( insert2 @ A @ X3 @ A6 ) )
= ( insert2 @ A @ X3 @ A6 ) ) ).
% insert_absorb2
thf(fact_64_insert__iff,axiom,
! [A: $tType,A3: A,B2: A,A6: set @ A] :
( ( member @ A @ A3 @ ( insert2 @ A @ B2 @ A6 ) )
= ( ( A3 = B2 )
| ( member @ A @ A3 @ A6 ) ) ) ).
% insert_iff
thf(fact_65_insertCI,axiom,
! [A: $tType,A3: A,B5: set @ A,B2: A] :
( ( ~ ( member @ A @ A3 @ B5 )
=> ( A3 = B2 ) )
=> ( member @ A @ A3 @ ( insert2 @ A @ B2 @ B5 ) ) ) ).
% insertCI
thf(fact_66_case4_I5_J,axiom,
( m
= ( suc @ na ) ) ).
% case4(5)
thf(fact_67_case4_I6_J,axiom,
( deg
= ( plus_plus @ nat @ na @ m ) ) ).
% case4(6)
thf(fact_68_ex__in__conv,axiom,
! [A: $tType,A6: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A6 ) )
= ( A6
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_69_equals0I,axiom,
! [A: $tType,A6: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A6 )
=> ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_70_equals0D,axiom,
! [A: $tType,A6: set @ A,A3: A] :
( ( A6
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A6 ) ) ).
% equals0D
thf(fact_71_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_72_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_73_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z: set @ A] : Y5 = Z )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_74_subset__trans,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% subset_trans
thf(fact_75_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_76_subset__refl,axiom,
! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ A6 ) ).
% subset_refl
thf(fact_77_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A7 )
=> ( member @ A @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_78_equalityD2,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( A6 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ).
% equalityD2
thf(fact_79_equalityD1,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( A6 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B5 ) ) ).
% equalityD1
thf(fact_80_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ A @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_81_equalityE,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( A6 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% equalityE
thf(fact_82_subsetD,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( member @ A @ C3 @ A6 )
=> ( member @ A @ C3 @ B5 ) ) ) ).
% subsetD
thf(fact_83_in__mono,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( member @ A @ X3 @ A6 )
=> ( member @ A @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_84_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A6: set @ A] :
( ( member @ A @ A3 @ A6 )
=> ? [B7: set @ A] :
( ( A6
= ( insert2 @ A @ A3 @ B7 ) )
& ~ ( member @ A @ A3 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_85_insert__commute,axiom,
! [A: $tType,X3: A,Y: A,A6: set @ A] :
( ( insert2 @ A @ X3 @ ( insert2 @ A @ Y @ A6 ) )
= ( insert2 @ A @ Y @ ( insert2 @ A @ X3 @ A6 ) ) ) ).
% insert_commute
thf(fact_86_insert__eq__iff,axiom,
! [A: $tType,A3: A,A6: set @ A,B2: A,B5: set @ A] :
( ~ ( member @ A @ A3 @ A6 )
=> ( ~ ( member @ A @ B2 @ B5 )
=> ( ( ( insert2 @ A @ A3 @ A6 )
= ( insert2 @ A @ B2 @ B5 ) )
= ( ( ( A3 = B2 )
=> ( A6 = B5 ) )
& ( ( A3 != B2 )
=> ? [C5: set @ A] :
( ( A6
= ( insert2 @ A @ B2 @ C5 ) )
& ~ ( member @ A @ B2 @ C5 )
& ( B5
= ( insert2 @ A @ A3 @ C5 ) )
& ~ ( member @ A @ A3 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_87_insert__absorb,axiom,
! [A: $tType,A3: A,A6: set @ A] :
( ( member @ A @ A3 @ A6 )
=> ( ( insert2 @ A @ A3 @ A6 )
= A6 ) ) ).
% insert_absorb
thf(fact_88_insert__ident,axiom,
! [A: $tType,X3: A,A6: set @ A,B5: set @ A] :
( ~ ( member @ A @ X3 @ A6 )
=> ( ~ ( member @ A @ X3 @ B5 )
=> ( ( ( insert2 @ A @ X3 @ A6 )
= ( insert2 @ A @ X3 @ B5 ) )
= ( A6 = B5 ) ) ) ) ).
% insert_ident
thf(fact_89_Set_Oset__insert,axiom,
! [A: $tType,X3: A,A6: set @ A] :
( ( member @ A @ X3 @ A6 )
=> ~ ! [B7: set @ A] :
( ( A6
= ( insert2 @ A @ X3 @ B7 ) )
=> ( member @ A @ X3 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_90_insertI2,axiom,
! [A: $tType,A3: A,B5: set @ A,B2: A] :
( ( member @ A @ A3 @ B5 )
=> ( member @ A @ A3 @ ( insert2 @ A @ B2 @ B5 ) ) ) ).
% insertI2
thf(fact_91_insertI1,axiom,
! [A: $tType,A3: A,B5: set @ A] : ( member @ A @ A3 @ ( insert2 @ A @ A3 @ B5 ) ) ).
% insertI1
thf(fact_92_insertE,axiom,
! [A: $tType,A3: A,B2: A,A6: set @ A] :
( ( member @ A @ A3 @ ( insert2 @ A @ B2 @ A6 ) )
=> ( ( A3 != B2 )
=> ( member @ A @ A3 @ A6 ) ) ) ).
% insertE
thf(fact_93_singleton__inject,axiom,
! [A: $tType,A3: A,B2: A] :
( ( ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B2 ) ) ).
% singleton_inject
thf(fact_94_insert__not__empty,axiom,
! [A: $tType,A3: A,A6: set @ A] :
( ( insert2 @ A @ A3 @ A6 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_95_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B2: A,C3: A,D3: A] :
( ( ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ C3 @ ( insert2 @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C3 )
& ( B2 = D3 ) )
| ( ( A3 = D3 )
& ( B2 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_96_singleton__iff,axiom,
! [A: $tType,B2: A,A3: A] :
( ( member @ A @ B2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A3 ) ) ).
% singleton_iff
thf(fact_97_singletonD,axiom,
! [A: $tType,B2: A,A3: A] :
( ( member @ A @ B2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A3 ) ) ).
% singletonD
thf(fact_98_subset__insertI2,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert2 @ A @ B2 @ B5 ) ) ) ).
% subset_insertI2
thf(fact_99_subset__insertI,axiom,
! [A: $tType,B5: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert2 @ A @ A3 @ B5 ) ) ).
% subset_insertI
thf(fact_100_subset__insert,axiom,
! [A: $tType,X3: A,A6: set @ A,B5: set @ A] :
( ~ ( member @ A @ X3 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A6 @ B5 ) ) ) ).
% subset_insert
thf(fact_101_insert__mono,axiom,
! [A: $tType,C4: set @ A,D4: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ C4 ) @ ( insert2 @ A @ A3 @ D4 ) ) ) ).
% insert_mono
thf(fact_102_buildup__nothing__in__leaf,axiom,
! [N: nat,X3: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X3 ) ).
% buildup_nothing_in_leaf
thf(fact_103_maxt__sound,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 )
=> ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X3 ) ) ) ) ).
% maxt_sound
thf(fact_104_maxt__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X3 ) )
=> ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 ) ) ) ).
% maxt_corr
thf(fact_105_mint__sound,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 )
=> ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X3 ) ) ) ) ).
% mint_sound
thf(fact_106_mint__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X3 ) )
=> ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 ) ) ) ).
% mint_corr
thf(fact_107_maxt__corr__help,axiom,
! [T2: vEBT_VEBT,N: nat,Maxi: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ Maxi ) )
=> ( ( vEBT_vebt_member @ T2 @ X3 )
=> ( ord_less_eq @ nat @ X3 @ Maxi ) ) ) ) ).
% maxt_corr_help
thf(fact_108_mint__corr__help,axiom,
! [T2: vEBT_VEBT,N: nat,Mini: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ Mini ) )
=> ( ( vEBT_vebt_member @ T2 @ X3 )
=> ( ord_less_eq @ nat @ Mini @ X3 ) ) ) ) ).
% mint_corr_help
thf(fact_109_maxt__member,axiom,
! [T2: vEBT_VEBT,N: nat,Maxi: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ Maxi ) )
=> ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ).
% maxt_member
thf(fact_110_mint__member,axiom,
! [T2: vEBT_VEBT,N: nat,Maxi: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ Maxi ) )
=> ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ).
% mint_member
thf(fact_111_the__elem__eq,axiom,
! [A: $tType,X3: A] :
( ( the_elem @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ).
% the_elem_eq
thf(fact_112_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_113_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_114_even__odd__cases,axiom,
! [X3: nat] :
( ! [N2: nat] :
( X3
!= ( plus_plus @ nat @ N2 @ N2 ) )
=> ~ ! [N2: nat] :
( X3
!= ( plus_plus @ nat @ N2 @ ( suc @ N2 ) ) ) ) ).
% even_odd_cases
thf(fact_115_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
=> ? [Info2: option @ ( product_prod @ nat @ nat ),TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S2 ) ) ) ).
% deg_SUcn_Node
thf(fact_116_maxbmo,axiom,
! [T2: vEBT_VEBT,X3: nat] :
( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X3 ) )
=> ( vEBT_V8194947554948674370ptions @ T2 @ X3 ) ) ).
% maxbmo
thf(fact_117_ac,axiom,
! [T2: vEBT_VEBT,H: nat,K2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ H )
=> ( ( vEBT_invar_vebt @ K2 @ H )
=> ( ( ( vEBT_VEBT_set_vebt @ T2 )
= ( vEBT_VEBT_set_vebt @ K2 ) )
=> ( ( vEBT_vebt_mint @ T2 )
= ( vEBT_vebt_mint @ K2 ) ) ) ) ) ).
% ac
thf(fact_118_ad,axiom,
! [T2: vEBT_VEBT,H: nat,K2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ H )
=> ( ( vEBT_invar_vebt @ K2 @ H )
=> ( ( ( vEBT_VEBT_set_vebt @ T2 )
= ( vEBT_VEBT_set_vebt @ K2 ) )
=> ( ( vEBT_vebt_maxt @ T2 )
= ( vEBT_vebt_maxt @ K2 ) ) ) ) ) ).
% ad
thf(fact_119_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_120_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_121_nle__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( ord_less_eq @ A @ A3 @ B2 ) )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( B2 != A3 ) ) ) ) ).
% nle_le
thf(fact_122_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_123_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_124_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( A3 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_125_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_126_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X3 )
=> ( X3 = Y ) ) ) ) ).
% order_antisym
thf(fact_127_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% order.trans
thf(fact_128_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_129_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: A,B4: A] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_130_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [A8: A,B8: A] :
( ( ord_less_eq @ A @ B8 @ A8 )
& ( ord_less_eq @ A @ A8 @ B8 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_131_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_132_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_133_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ) ).
% antisym
thf(fact_134_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) ) ) ).
% le_funD
thf(fact_135_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) ) ) ).
% le_funE
thf(fact_136_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B] :
( ! [X5: A] : ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G3 ) ) ) ).
% le_funI
thf(fact_137_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_138_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [A8: A,B8: A] :
( ( ord_less_eq @ A @ A8 @ B8 )
& ( ord_less_eq @ A @ B8 @ A8 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_139_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F3: B > A,B2: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less_eq @ B @ X5 @ Y4 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_140_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F3: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ C @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_141_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( X3 = Y )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% order_eq_refl
thf(fact_142_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
| ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% linorder_linear
thf(fact_143_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F3: B > A,B2: B,C3: B] :
( ( A3
= ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less_eq @ B @ X5 @ Y4 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_144_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F3: A > B,C3: B] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( F3 @ B2 )
= C3 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_145_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% linorder_le_cases
thf(fact_146_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
= ( X3 = Y ) ) ) ) ).
% order_antisym_conv
thf(fact_147_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_148_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_149_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_150_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_151_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ Tree @ N )
=> ( ( vEBT_vebt_member @ Tree @ X3 )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X3 )
| ( vEBT_VEBT_membermima @ Tree @ X3 ) ) ) ) ).
% member_valid_both_member_options
thf(fact_152_vebt__maxt_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( some @ nat @ Ma ) ) ).
% vebt_maxt.simps(3)
thf(fact_153_vebt__mint_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( some @ nat @ Mi ) ) ).
% vebt_mint.simps(3)
thf(fact_154_nat__add__left__cancel__le,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K2 @ M2 ) @ ( plus_plus @ nat @ K2 @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_155_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus @ nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_156_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq @ nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_157_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_158_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_159_add__shift,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ( plus_plus @ nat @ X3 @ Y )
= Z2 )
= ( ( vEBT_VEBT_add @ ( some @ nat @ X3 ) @ ( some @ nat @ Y ) )
= ( some @ nat @ Z2 ) ) ) ).
% add_shift
thf(fact_160_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T3: vEBT_VEBT,X4: nat] :
( ( vEBT_V5719532721284313246member @ T3 @ X4 )
| ( vEBT_VEBT_membermima @ T3 @ X4 ) ) ) ) ).
% both_member_options_def
thf(fact_161_maxt__corr__help__empty,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( none @ nat ) )
=> ( ( vEBT_VEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% maxt_corr_help_empty
thf(fact_162_mint__corr__help__empty,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) )
=> ( ( vEBT_VEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mint_corr_help_empty
thf(fact_163_buildup__nothing__in__min__max,axiom,
! [N: nat,X3: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X3 ) ).
% buildup_nothing_in_min_max
thf(fact_164_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
= ( B2 = C3 ) ) ) ).
% add_right_cancel
thf(fact_165_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C3 ) )
= ( B2 = C3 ) ) ) ).
% add_left_cancel
thf(fact_166_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_167_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_168_not__None__eq,axiom,
! [A: $tType,X3: option @ A] :
( ( X3
!= ( none @ A ) )
= ( ? [Y3: A] :
( X3
= ( some @ A @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_169_not__Some__eq,axiom,
! [A: $tType,X3: option @ A] :
( ( ! [Y3: A] :
( X3
!= ( some @ A @ Y3 ) ) )
= ( X3
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_170_add__def,axiom,
( vEBT_VEBT_add
= ( vEBT_V2048590022279873568_shift @ nat @ ( plus_plus @ nat ) ) ) ).
% add_def
thf(fact_171_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
( ! [Uu: A > A > $o,Uv: option @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv ) ) )
=> ( ! [Uw: A > A > $o,V: A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V ) @ ( none @ A ) ) ) )
=> ~ ! [F2: A > A > $o,X5: A,Y4: A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ X5 ) @ ( some @ A @ Y4 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.cases
thf(fact_172_VEBT__internal_Ooption__shift_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
( ! [Uu: A > A > A,Uv: option @ A] :
( X3
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv ) ) )
=> ( ! [Uw: A > A > A,V: A] :
( X3
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V ) @ ( none @ A ) ) ) )
=> ~ ! [F2: A > A > A,A5: A,B4: A] :
( X3
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ A5 ) @ ( some @ A @ B4 ) ) ) ) ) ) ).
% VEBT_internal.option_shift.cases
thf(fact_173_option_Odistinct_I1_J,axiom,
! [A: $tType,X2: A] :
( ( none @ A )
!= ( some @ A @ X2 ) ) ).
% option.distinct(1)
thf(fact_174_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X2: A] :
( ( Option
= ( some @ A @ X2 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_175_option_Oexhaust,axiom,
! [A: $tType,Y: option @ A] :
( ( Y
!= ( none @ A ) )
=> ~ ! [X22: A] :
( Y
!= ( some @ A @ X22 ) ) ) ).
% option.exhaust
thf(fact_176_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P3: ( option @ A ) > $o] :
? [X7: option @ A] : ( P3 @ X7 ) )
= ( ^ [P4: ( option @ A ) > $o] :
( ( P4 @ ( none @ A ) )
| ? [X4: A] : ( P4 @ ( some @ A @ X4 ) ) ) ) ) ).
% split_option_ex
thf(fact_177_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P3: ( option @ A ) > $o] :
! [X7: option @ A] : ( P3 @ X7 ) )
= ( ^ [P4: ( option @ A ) > $o] :
( ( P4 @ ( none @ A ) )
& ! [X4: A] : ( P4 @ ( some @ A @ X4 ) ) ) ) ) ).
% split_option_all
thf(fact_178_combine__options__cases,axiom,
! [A: $tType,B: $tType,X3: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] :
( ( ( X3
= ( none @ A ) )
=> ( P @ X3 @ Y ) )
=> ( ( ( Y
= ( none @ B ) )
=> ( P @ X3 @ Y ) )
=> ( ! [A5: A,B4: B] :
( ( X3
= ( some @ A @ A5 ) )
=> ( ( Y
= ( some @ B @ B4 ) )
=> ( P @ X3 @ Y ) ) )
=> ( P @ X3 @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_179_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
=> ( B2 = C3 ) ) ) ).
% add_right_imp_eq
thf(fact_180_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C3 ) )
=> ( B2 = C3 ) ) ) ).
% add_left_imp_eq
thf(fact_181_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A3 @ C3 ) )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add.left_commute
thf(fact_182_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A8: A,B8: A] : ( plus_plus @ A @ B8 @ A8 ) ) ) ) ).
% add.commute
thf(fact_183_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
= ( B2 = C3 ) ) ) ).
% add.right_cancel
thf(fact_184_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C3 ) )
= ( B2 = C3 ) ) ) ).
% add.left_cancel
thf(fact_185_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add.assoc
thf(fact_186_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B5: A,K2: A,B2: A,A3: A] :
( ( B5
= ( plus_plus @ A @ K2 @ B2 ) )
=> ( ( plus_plus @ A @ A3 @ B5 )
= ( plus_plus @ A @ K2 @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_187_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: A,K2: A,A3: A,B2: A] :
( ( A6
= ( plus_plus @ A @ K2 @ A3 ) )
=> ( ( plus_plus @ A @ A6 @ B2 )
= ( plus_plus @ A @ K2 @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_188_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus @ A @ I @ K2 )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_189_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_190_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_191_Suc__inject,axiom,
! [X3: nat,Y: nat] :
( ( ( suc @ X3 )
= ( suc @ Y ) )
=> ( X3 = Y ) ) ).
% Suc_inject
thf(fact_192_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B2: nat] :
( ( P @ K2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq @ nat @ Y6 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_193_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
| ( ord_less_eq @ nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_194_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_195_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_196_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K2 )
=> ( ord_less_eq @ nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_197_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_198_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X3: A,Y: A] :
( ( ( size_size @ A @ X3 )
!= ( size_size @ A @ Y ) )
=> ( X3 != Y ) ) ) ).
% size_neq_size_imp_neq
thf(fact_199_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_200_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_201_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] :
? [C6: A] :
( B8
= ( plus_plus @ A @ A8 @ C6 ) ) ) ) ) ).
% le_iff_add
thf(fact_202_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add_right_mono
thf(fact_203_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ! [C2: A] :
( B2
!= ( plus_plus @ A @ A3 @ C2 ) ) ) ) ).
% less_eqE
thf(fact_204_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_205_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_mono
thf(fact_206_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_207_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_208_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_209_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ! [X5: nat] : ( R @ X5 @ X5 )
=> ( ! [X5: nat,Y4: nat,Z3: nat] :
( ( R @ X5 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X5 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_210_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_211_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_212_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq @ nat @ M2 @ N ) )
= ( ord_less_eq @ nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_213_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_214_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq @ nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_215_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M4 )
=> ? [M: nat] :
( M4
= ( suc @ M ) ) ) ).
% Suc_le_D
thf(fact_216_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_217_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq @ nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_218_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_219_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M2 ) @ N )
= ( plus_plus @ nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_220_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_221_nat__arith_Osuc1,axiom,
! [A6: nat,K2: nat,A3: nat] :
( ( A6
= ( plus_plus @ nat @ K2 @ A3 ) )
=> ( ( suc @ A6 )
= ( plus_plus @ nat @ K2 @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_222_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M5: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus @ nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_223_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_224_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_225_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K2 ) @ ( plus_plus @ nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_226_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K2 @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K2 ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_227_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq @ nat @ K2 @ L )
=> ? [N2: nat] :
( L
= ( plus_plus @ nat @ K2 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_228_add__leD2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ N )
=> ( ord_less_eq @ nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_229_add__leD1,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_230_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M2 @ N ) ) ).
% le_add2
thf(fact_231_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M2 ) ) ).
% le_add1
thf(fact_232_add__leE,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ N )
=> ~ ( ( ord_less_eq @ nat @ M2 @ N )
=> ~ ( ord_less_eq @ nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_233_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ord_less_eq @ A @ ( F3 @ N4 ) @ ( F3 @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_234_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ord_less_eq @ A @ ( F3 @ N ) @ ( F3 @ N4 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_235_geqmaxNone,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X3: 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 @ Ma @ X3 )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( none @ nat ) ) ) ) ).
% geqmaxNone
thf(fact_236_minminNull,axiom,
! [T2: vEBT_VEBT] :
( ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) )
=> ( vEBT_VEBT_minNull @ T2 ) ) ).
% minminNull
thf(fact_237_minNullmin,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ T2 )
=> ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) ) ) ).
% minNullmin
thf(fact_238_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_239_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_240_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
( A7
= ( insert2 @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_241_valid__eq2,axiom,
! [T2: vEBT_VEBT,D3: nat] :
( ( vEBT_VEBT_valid @ T2 @ D3 )
=> ( vEBT_invar_vebt @ T2 @ D3 ) ) ).
% valid_eq2
thf(fact_242_valid__eq1,axiom,
! [T2: vEBT_VEBT,D3: nat] :
( ( vEBT_invar_vebt @ T2 @ D3 )
=> ( vEBT_VEBT_valid @ T2 @ D3 ) ) ).
% valid_eq1
thf(fact_243_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_244_sprop1,axiom,
( ( sa
= ( vEBT_Node @ info @ deg @ treeList @ summary ) )
& ( deg
= ( plus_plus @ nat @ na @ m ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ treeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
& ( vEBT_invar_vebt @ summary @ m )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ treeList ) )
=> ( vEBT_invar_vebt @ X @ na ) ) ) ).
% sprop1
thf(fact_245_is__singletonI,axiom,
! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_246_not__min__Null__member,axiom,
! [T2: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ T2 )
=> ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_12 ) ) ).
% not_min_Null_member
thf(fact_247_min__Null__member,axiom,
! [T2: vEBT_VEBT,X3: nat] :
( ( vEBT_VEBT_minNull @ T2 )
=> ~ ( vEBT_vebt_member @ T2 @ X3 ) ) ).
% min_Null_member
thf(fact_248_power__shift,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ( power_power @ nat @ X3 @ Y )
= Z2 )
= ( ( vEBT_VEBT_power @ ( some @ nat @ X3 ) @ ( some @ nat @ Y ) )
= ( some @ nat @ Z2 ) ) ) ).
% power_shift
thf(fact_249_case4_I4_J,axiom,
( ( size_size @ ( list @ vEBT_VEBT ) @ treeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) ) ).
% case4(4)
thf(fact_250_local_Opower__def,axiom,
( vEBT_VEBT_power
= ( vEBT_V2048590022279873568_shift @ nat @ ( power_power @ nat ) ) ) ).
% local.power_def
thf(fact_251_a0,axiom,
ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg ).
% a0
thf(fact_252__092_060open_062length_AtreeList_H_A_061_A2_A_094_Am_092_060close_062,axiom,
( ( size_size @ ( list @ vEBT_VEBT ) @ treeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) ) ).
% \<open>length treeList' = 2 ^ m\<close>
thf(fact_253__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062treeList_H_Asummary_H_Ainfo_O_As_A_061_ANode_Ainfo_Adeg_AtreeList_H_Asummary_H_A_092_060and_062_Adeg_A_061_An_A_L_Am_A_092_060and_062_Alength_AtreeList_H_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_H_Am_A_092_060and_062_A_I_092_060forall_062t_092_060in_062set_AtreeList_H_O_Ainvar__vebt_At_An_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Info2: option @ ( product_prod @ nat @ nat )] :
~ ( ( sa
= ( vEBT_Node @ Info2 @ deg @ TreeList3 @ Summary2 ) )
& ( deg
= ( plus_plus @ nat @ na @ m ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
& ( vEBT_invar_vebt @ Summary2 @ m )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X @ na ) ) ) ).
% \<open>\<And>thesis. (\<And>treeList' summary' info. s = Node info deg treeList' summary' \<and> deg = n + m \<and> length treeList' = 2 ^ m \<and> invar_vebt summary' m \<and> (\<forall>t\<in>set treeList'. invar_vebt t n) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_254_succ__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X3 )
= ( some @ nat @ Sx ) )
= ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 @ Sx ) ) ) ).
% succ_corr
thf(fact_255_succ__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X3 )
= ( some @ nat @ Sx ) )
= ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T2 ) @ X3 @ Sx ) ) ) ).
% succ_correct
thf(fact_256_case4_I10_J,axiom,
ord_less @ nat @ ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg ) ).
% case4(10)
thf(fact_257_invar__vebt_Ointros_I2_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2 = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M2 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_258_invar__vebt_Ointros_I3_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M2 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_259_is__singletonI_H,axiom,
! [A: $tType,A6: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ A6 )
=> ( ( member @ A @ Y4 @ A6 )
=> ( X5 = Y4 ) ) )
=> ( is_singleton @ A @ A6 ) ) ) ).
% is_singletonI'
thf(fact_260_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [A: $tType,F3: A > A > A,A3: A,B2: A] :
( ( vEBT_V2048590022279873568_shift @ A @ F3 @ ( some @ A @ A3 ) @ ( some @ A @ B2 ) )
= ( some @ A @ ( F3 @ A3 @ B2 ) ) ) ).
% VEBT_internal.option_shift.simps(3)
thf(fact_261_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
! [A: $tType,Uu2: A > A > A,Uv2: option @ A] :
( ( vEBT_V2048590022279873568_shift @ A @ Uu2 @ ( none @ A ) @ Uv2 )
= ( none @ A ) ) ).
% VEBT_internal.option_shift.simps(1)
thf(fact_262_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X5: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ).
% subrelI
thf(fact_263_vebt__mint_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
= ( none @ nat ) ) ).
% vebt_mint.simps(2)
thf(fact_264_vebt__maxt_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
= ( none @ nat ) ) ).
% vebt_maxt.simps(2)
thf(fact_265_is__singletonE,axiom,
! [A: $tType,A6: set @ A] :
( ( is_singleton @ A @ A6 )
=> ~ ! [X5: A] :
( A6
!= ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_266_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
? [X4: A] :
( A7
= ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_267_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [A: $tType,Uw2: A > A > A,V2: A] :
( ( vEBT_V2048590022279873568_shift @ A @ Uw2 @ ( some @ A @ V2 ) @ ( none @ A ) )
= ( none @ A ) ) ).
% VEBT_internal.option_shift.simps(2)
thf(fact_268_VEBT__internal_Ooption__shift_Oelims,axiom,
! [A: $tType,X3: A > A > A,Xa2: option @ A,Xb: option @ A,Y: option @ A] :
( ( ( vEBT_V2048590022279873568_shift @ A @ X3 @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2
= ( none @ A ) )
=> ( Y
!= ( none @ A ) ) )
=> ( ( ? [V: A] :
( Xa2
= ( some @ A @ V ) )
=> ( ( Xb
= ( none @ A ) )
=> ( Y
!= ( none @ A ) ) ) )
=> ~ ! [A5: A] :
( ( Xa2
= ( some @ A @ A5 ) )
=> ! [B4: A] :
( ( Xb
= ( some @ A @ B4 ) )
=> ( Y
!= ( some @ A @ ( X3 @ A5 @ B4 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_269_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_270_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_271_insert__simp__mima,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X3 = Mi )
| ( X3 = 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 ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% insert_simp_mima
thf(fact_272_succ__min,axiom,
! [Deg: nat,X3: nat,Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less @ nat @ X3 @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( some @ nat @ Mi ) ) ) ) ).
% succ_min
thf(fact_273_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_274_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% Suc_numeral
thf(fact_275_set__n__deg__not__0,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,M2: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N ) )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ N ) ) ) ).
% set_n_deg_not_0
thf(fact_276_del__single__cont,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X3 = Mi )
& ( X3 = Ma ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% del_single_cont
thf(fact_277_helpyd,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X3 )
= ( some @ nat @ Y ) )
=> ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% helpyd
thf(fact_278_misiz,axiom,
! [T2: vEBT_VEBT,N: nat,M2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( some @ nat @ M2 )
= ( vEBT_vebt_mint @ T2 ) )
=> ( ord_less @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% misiz
thf(fact_279_member__bound,axiom,
! [Tree: vEBT_VEBT,X3: nat,N: nat] :
( ( vEBT_vebt_member @ Tree @ X3 )
=> ( ( vEBT_invar_vebt @ Tree @ N )
=> ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% member_bound
thf(fact_280_delete__pres__valid,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T2 @ X3 ) @ N ) ) ).
% delete_pres_valid
thf(fact_281_dele__bmo__cont__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T2 @ X3 ) @ Y )
= ( ( X3 != Y )
& ( vEBT_V8194947554948674370ptions @ T2 @ Y ) ) ) ) ).
% dele_bmo_cont_corr
thf(fact_282_dele__member__cont__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T2 @ X3 ) @ Y )
= ( ( X3 != Y )
& ( vEBT_vebt_member @ T2 @ Y ) ) ) ) ).
% dele_member_cont_corr
thf(fact_283_numeral__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M2: num,N: num] :
( ( ( numeral_numeral @ A @ M2 )
= ( numeral_numeral @ A @ N ) )
= ( M2 = N ) ) ) ).
% numeral_eq_iff
thf(fact_284_succ__member,axiom,
! [T2: vEBT_VEBT,X3: nat,Y: nat] :
( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 @ Y )
= ( ( vEBT_vebt_member @ T2 @ Y )
& ( ord_less @ nat @ X3 @ Y )
& ! [Z4: nat] :
( ( ( vEBT_vebt_member @ T2 @ Z4 )
& ( ord_less @ nat @ X3 @ Z4 ) )
=> ( ord_less_eq @ nat @ Y @ Z4 ) ) ) ) ).
% succ_member
thf(fact_285_valid__pres__insert,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ T2 @ X3 ) @ N ) ) ) ).
% valid_pres_insert
thf(fact_286_numeral__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: num,N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ num @ M2 @ N ) ) ) ).
% numeral_le_iff
thf(fact_287_numeral__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: num,N: num] :
( ( ord_less @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ num @ M2 @ N ) ) ) ).
% numeral_less_iff
thf(fact_288_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_289_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_290_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_291_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_292_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_293_valid__insert__both__member__options__add,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ X3 ) @ X3 ) ) ) ).
% valid_insert_both_member_options_add
thf(fact_294_valid__insert__both__member__options__pres,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X3 @ ( 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 @ X3 )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ Y ) @ X3 ) ) ) ) ) ).
% valid_insert_both_member_options_pres
thf(fact_295_nat__add__left__cancel__less,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K2 @ M2 ) @ ( plus_plus @ nat @ K2 @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_296_post__member__pre__member,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X3 @ ( 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 @ X3 ) @ Y )
=> ( ( vEBT_vebt_member @ T2 @ Y )
| ( X3 = Y ) ) ) ) ) ) ).
% post_member_pre_member
thf(fact_297_delt__out__of__range,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X3 @ Mi )
| ( ord_less @ nat @ Ma @ X3 ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% delt_out_of_range
thf(fact_298_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_299_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_300_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [V2: num,W: num,Z2: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Z2 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W ) ) @ Z2 ) ) ) ).
% add_numeral_left
thf(fact_301_numeral__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [M2: num,N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ).
% numeral_plus_numeral
thf(fact_302_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_303_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_304_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_305_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_306_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_307_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_308_pred__member,axiom,
! [T2: vEBT_VEBT,X3: nat,Y: nat] :
( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 @ Y )
= ( ( vEBT_vebt_member @ T2 @ Y )
& ( ord_less @ nat @ Y @ X3 )
& ! [Z4: nat] :
( ( ( vEBT_vebt_member @ T2 @ Z4 )
& ( ord_less @ nat @ Z4 @ X3 ) )
=> ( ord_less_eq @ nat @ Z4 @ Y ) ) ) ) ).
% pred_member
thf(fact_309_case4_I7_J,axiom,
! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList2 @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ summary2 @ I2 ) ) ) ).
% case4(7)
thf(fact_310_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_311_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X3: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ).
% lt_ex
thf(fact_312_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X3: A] :
? [X_12: A] : ( ord_less @ A @ X3 @ X_12 ) ) ).
% gt_ex
thf(fact_313_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X3 @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_314_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( X3 != Y ) ) ) ).
% less_imp_neq
thf(fact_315_order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% order.asym
thf(fact_316_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( A3 = B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_317_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_318_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X5: A] :
( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X5 )
=> ( P @ Y6 ) )
=> ( P @ X5 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_319_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X3: A] :
( ~ ( ord_less @ A @ Y @ X3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( X3 = Y ) ) ) ) ).
% antisym_conv3
thf(fact_320_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% linorder_cases
thf(fact_321_dual__order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_322_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_323_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P3: A > $o] :
? [X7: A] : ( P3 @ X7 ) )
= ( ^ [P4: A > $o] :
? [N3: A] :
( ( P4 @ N3 )
& ! [M5: A] :
( ( ord_less @ A @ M5 @ N3 )
=> ~ ( P4 @ M5 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_324_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: A] : ( P @ A5 @ A5 )
=> ( ! [A5: A,B4: A] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A3 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_325_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% order.strict_trans
thf(fact_326_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( ( ord_less @ A @ Y @ X3 )
| ( X3 = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_327_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_328_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( A3 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_329_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( A3 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_330_linorder__neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
=> ( ~ ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% linorder_neqE
thf(fact_331_order__less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% order_less_asym
thf(fact_332_linorder__neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
= ( ( ord_less @ A @ X3 @ Y )
| ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% linorder_neq_iff
thf(fact_333_order__less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% order_less_asym'
thf(fact_334_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% order_less_trans
thf(fact_335_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F3: B > A,B2: B,C3: B] :
( ( A3
= ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less @ B @ X5 @ Y4 )
=> ( ord_less @ A @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_336_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F3: A > B,C3: B] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ( F3 @ B2 )
= C3 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less @ A @ X5 @ Y4 )
=> ( ord_less @ B @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ B @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_337_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 ) ) ).
% order_less_irrefl
thf(fact_338_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F3: B > A,B2: B,C3: B] :
( ( ord_less @ A @ A3 @ ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less @ B @ X5 @ Y4 )
=> ( ord_less @ A @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_339_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F3: A > C,C3: C] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ C @ ( F3 @ B2 ) @ C3 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less @ A @ X5 @ Y4 )
=> ( ord_less @ C @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ C @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% order_less_subst2
thf(fact_340_order__less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% order_less_not_sym
thf(fact_341_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,P: $o] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less @ A @ Y @ X3 )
=> P ) ) ) ).
% order_less_imp_triv
thf(fact_342_linorder__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
| ( X3 = Y )
| ( ord_less @ A @ Y @ X3 ) ) ) ).
% linorder_less_linear
thf(fact_343_order__less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( X3 != Y ) ) ) ).
% order_less_imp_not_eq
thf(fact_344_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( Y != X3 ) ) ) ).
% order_less_imp_not_eq2
thf(fact_345_order__less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% order_less_imp_not_less
thf(fact_346_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less @ nat @ M2 @ N )
| ( ord_less @ nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_347_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_348_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_349_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_350_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P: A > $o,A3: A] :
( ! [X5: A] :
( ! [Y6: A] :
( ( ord_less @ B @ ( F3 @ Y6 ) @ ( F3 @ X5 ) )
=> ( P @ Y6 ) )
=> ( P @ X5 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct
thf(fact_351_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X3: A] :
( ( ( one_one @ A )
= X3 )
= ( X3
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_352_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_353_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_354_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_355_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less @ nat @ X3 @ Y )
=> ( ord_less @ nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_356_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P: A > $o,A3: A] :
( ! [X5: A] :
( ! [Y6: A] :
( ( ord_less @ B @ ( F3 @ Y6 ) @ ( F3 @ X5 ) )
=> ( P @ Y6 ) )
=> ( P @ X5 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct_rule
thf(fact_357_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V3: A > nat,X3: A] :
( ! [X5: A] :
( ~ ( P @ X5 )
=> ? [Y6: A] :
( ( ord_less @ nat @ ( V3 @ Y6 ) @ ( V3 @ X5 ) )
& ~ ( P @ Y6 ) ) )
=> ( P @ X3 ) ) ).
% infinite_descent_measure
thf(fact_358_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N4 )
=> ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ N4 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_359_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ M2 ) )
= ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_360_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_361_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less @ A @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_362_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
| ( ord_less @ A @ Y @ X3 ) ) ) ).
% linorder_le_less_linear
thf(fact_363_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F3: A > C,C3: C] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less @ A @ X5 @ Y4 )
=> ( ord_less @ C @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ C @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_364_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F3: B > A,B2: B,C3: B] :
( ( ord_less @ A @ A3 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less_eq @ B @ X5 @ Y4 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_365_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F3: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ C @ ( F3 @ B2 ) @ C3 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ C @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ C @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_366_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F3: B > A,B2: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less @ B @ X5 @ Y4 )
=> ( ord_less @ A @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_367_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% order_less_le_trans
thf(fact_368_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% order_le_less_trans
thf(fact_369_order__neq__le__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( A3 != B2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% order_neq_le_trans
thf(fact_370_order__le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% order_le_neq_trans
thf(fact_371_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% order_less_imp_le
thf(fact_372_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% linorder_not_less
thf(fact_373_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X3 @ Y ) )
= ( ord_less @ A @ Y @ X3 ) ) ) ).
% linorder_not_le
thf(fact_374_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ) ).
% order_less_le
thf(fact_375_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ) ).
% order_le_less
thf(fact_376_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_377_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_378_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B8: A,A8: A] :
( ( ord_less_eq @ A @ B8 @ A8 )
& ~ ( ord_less_eq @ A @ A8 @ B8 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_379_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_380_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_381_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B8: A,A8: A] :
( ( ord_less_eq @ A @ B8 @ A8 )
& ( A8 != B8 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_382_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B8: A,A8: A] :
( ( ord_less @ A @ B8 @ A8 )
| ( A8 = B8 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_383_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ! [W2: A] :
( ( ord_less @ A @ X3 @ W2 )
=> ( ( ord_less @ A @ W2 @ Y )
=> ( ord_less_eq @ A @ W2 @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_384_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ! [W2: A] :
( ( ord_less @ A @ Z2 @ W2 )
=> ( ( ord_less @ A @ W2 @ X3 )
=> ( ord_less_eq @ A @ Y @ W2 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_385_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A8: A,B8: A] :
( ( ord_less_eq @ A @ A8 @ B8 )
& ~ ( ord_less_eq @ A @ B8 @ A8 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_386_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% order.strict_trans2
thf(fact_387_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% order.strict_trans1
thf(fact_388_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A8: A,B8: A] :
( ( ord_less_eq @ A @ A8 @ B8 )
& ( A8 != B8 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_389_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] :
( ( ord_less @ A @ A8 @ B8 )
| ( A8 = B8 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_390_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X3: A] :
( ~ ( ord_less_eq @ A @ Y @ X3 )
=> ( ord_less @ A @ X3 @ Y ) ) ) ).
% not_le_imp_less
thf(fact_391_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
& ~ ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% less_le_not_le
thf(fact_392_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z2: A] :
( ! [X5: A] :
( ( ord_less @ A @ X5 @ Y )
=> ( ord_less_eq @ A @ X5 @ Z2 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_le
thf(fact_393_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,Y: A] :
( ! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( ord_less_eq @ A @ Y @ X5 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_ge
thf(fact_394_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( X3 = Y ) ) ) ) ).
% antisym_conv2
thf(fact_395_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ X3 @ Y )
= ( X3 = Y ) ) ) ) ).
% antisym_conv1
thf(fact_396_nless__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( ord_less @ A @ A3 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% nless_le
thf(fact_397_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% leI
thf(fact_398_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ~ ( ord_less @ A @ X3 @ Y ) ) ) ).
% leD
thf(fact_399_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_400_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B2 ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_401_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add_strict_right_mono
thf(fact_402_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_403_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_strict_mono
thf(fact_404_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K2 = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_405_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_406_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_407_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_408_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_409_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_410_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_411_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K )
=> ( P @ I3 @ K ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_412_less__trans__Suc,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ K2 )
=> ( ord_less @ nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_413_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_414_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_415_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less @ nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_416_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_417_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M2 @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_418_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
= ( ( ord_less @ nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_419_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_420_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_421_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_422_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less @ nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_423_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_424_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ N )
=> ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_425_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less @ nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_426_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ord_less @ nat @ ( F3 @ I3 ) @ ( F3 @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F3 @ I ) @ ( F3 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_427_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_428_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less @ nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_429_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M5: nat,N3: nat] :
( ( ord_less @ nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_430_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_431_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_432_less__add__eq__less,axiom,
! [K2: nat,L: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ K2 @ L )
=> ( ( ( plus_plus @ nat @ M2 @ L )
= ( plus_plus @ nat @ K2 @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_433_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_434_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_435_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K2 ) @ ( plus_plus @ nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_436_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_437_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_438_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ K2 @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K2 ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_439_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K2 )
=> ( ord_less @ nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_440_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_441_one__plus__numeral__commute,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X3: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X3 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X3 ) @ ( one_one @ A ) ) ) ) ).
% one_plus_numeral_commute
thf(fact_442_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_443_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_444_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_445_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_446_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_447_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_448_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less @ nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_449_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N3: nat] : ( ord_less_eq @ nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_450_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_451_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_452_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
=> ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_453_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( ord_less @ nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_454_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( ord_less @ nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_455_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
= ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_456_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_457_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ? [K: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M2 @ K ) ) ) ) ).
% less_imp_Suc_add
thf(fact_458_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M5: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus @ nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_459_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_460_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_461_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M2 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_462_mono__nat__linear__lb,axiom,
! [F3: nat > nat,M2: nat,K2: nat] :
( ! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ nat @ ( F3 @ M ) @ ( F3 @ N2 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F3 @ M2 ) @ K2 ) @ ( F3 @ ( plus_plus @ nat @ M2 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_463_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_464_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_465_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_466_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_467_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_468_add__One__commute,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ N )
= ( plus_plus @ num @ N @ one2 ) ) ).
% add_One_commute
thf(fact_469_le__num__One__iff,axiom,
! [X3: num] :
( ( ord_less_eq @ num @ X3 @ one2 )
= ( X3 = one2 ) ) ).
% le_num_One_iff
thf(fact_470_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_471_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_472_pred__max,axiom,
! [Deg: nat,Ma: nat,X3: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less @ nat @ Ma @ X3 )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( some @ nat @ Ma ) ) ) ) ).
% pred_max
thf(fact_473_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X3: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X3 ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less_eq @ nat @ X3 @ Y ) ) ) ) ).
% power_increasing_iff
thf(fact_474_greater__shift,axiom,
( ( ord_less @ nat )
= ( ^ [Y3: nat,X4: nat] : ( vEBT_VEBT_greater @ ( some @ nat @ X4 ) @ ( some @ nat @ Y3 ) ) ) ) ).
% greater_shift
thf(fact_475_less__shift,axiom,
( ( ord_less @ nat )
= ( ^ [X4: nat,Y3: nat] : ( vEBT_VEBT_less @ ( some @ nat @ X4 ) @ ( some @ nat @ Y3 ) ) ) ) ).
% less_shift
thf(fact_476_helpypredd,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X3 )
= ( some @ nat @ Y ) )
=> ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% helpypredd
thf(fact_477_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X3: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ X3 ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less @ nat @ X3 @ Y ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_478_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
! [Deg: nat,X3: nat,Info: option @ ( product_prod @ nat @ nat ),TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) @ X3 )
=> ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) )
& ( ~ ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) @ X3 )
=> ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X3 ) ) ) ) ).
% VEBT_internal.insert'.simps(2)
thf(fact_479_sumtreelistcong,axiom,
! [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 ) ) ) ).
% sumtreelistcong
thf(fact_480_setcongy,axiom,
! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( vEBT_VEBT_set_vebt @ ( nth @ vEBT_VEBT @ treeList2 @ I ) )
= ( vEBT_VEBT_set_vebt @ ( nth @ vEBT_VEBT @ treeList @ I ) ) ) ) ).
% setcongy
thf(fact_481_membercongy,axiom,
! [I: nat,X3: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ treeList2 @ I ) @ X3 )
= ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ treeList @ I ) @ X3 ) ) ) ).
% membercongy
thf(fact_482_ex__power__ivl2,axiom,
! [B2: nat,K2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K2 )
& ( ord_less_eq @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_483_inthall,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o,N: nat] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X5 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_484_psubsetI,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( A6 != B5 )
=> ( ord_less @ ( set @ A ) @ A6 @ B5 ) ) ) ).
% psubsetI
thf(fact_485_pred__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Px: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X3 )
= ( some @ nat @ Px ) )
= ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 @ Px ) ) ) ).
% pred_corr
thf(fact_486_pred__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X3 )
= ( some @ nat @ Sx ) )
= ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T2 ) @ X3 @ Sx ) ) ) ).
% pred_correct
thf(fact_487_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_488_power__one__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% power_one_right
thf(fact_489_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ( power_power @ A @ A3 @ M2 )
= ( power_power @ A @ A3 @ N ) )
= ( M2 = N ) ) ) ) ).
% power_inject_exp
thf(fact_490_treecongy,axiom,
! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( nth @ vEBT_VEBT @ treeList2 @ I )
= ( nth @ vEBT_VEBT @ treeList @ I ) ) ) ).
% treecongy
thf(fact_491_not__psubset__empty,axiom,
! [A: $tType,A6: set @ A] :
~ ( ord_less @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_492_psubsetE,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% psubsetE
thf(fact_493_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
& ( A7 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_494_psubset__imp__subset,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_495_psubset__subset__trans,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_496_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
& ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_497_subset__psubset__trans,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( ord_less @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_498_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B6 )
| ( A7 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_499_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
( ( ord_less_eq @ ( A > B ) @ F4 @ G4 )
& ~ ( ord_less_eq @ ( A > B ) @ G4 @ F4 ) ) ) ) ) ).
% less_fun_def
thf(fact_500_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_501_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_502_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N5: nat,A3: A] :
( ( ord_less @ nat @ N @ N5 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N5 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_503_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_504_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N5: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N5 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N5 ) ) ) ) ) ).
% power_increasing
thf(fact_505_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_506_one__power2,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( power_power @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% one_power2
thf(fact_507_less__exp,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% less_exp
thf(fact_508_self__le__ge2__pow,axiom,
! [K2: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ( ord_less_eq @ nat @ M2 @ ( power_power @ nat @ K2 @ M2 ) ) ) ).
% self_le_ge2_pow
thf(fact_509_power2__nat__le__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_510_power2__nat__le__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_511_ex__power__ivl1,axiom,
! [B2: nat,K2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K2 )
=> ? [N2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K2 )
& ( ord_less @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_512_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_513_semiring__norm_I69_J,axiom,
! [M2: num] :
~ ( ord_less_eq @ num @ ( bit0 @ M2 ) @ one2 ) ).
% semiring_norm(69)
thf(fact_514_insert__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( sup_sup @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert2 @ nat @ X3 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( vEBT_set_vebt @ ( vEBT_vebt_insert @ T2 @ X3 ) ) ) ) ) ).
% insert_correct
thf(fact_515_semiring__norm_I2_J,axiom,
( ( plus_plus @ num @ one2 @ one2 )
= ( bit0 @ one2 ) ) ).
% semiring_norm(2)
thf(fact_516_insert__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( sup_sup @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( insert2 @ nat @ X3 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_insert @ T2 @ X3 ) ) ) ) ) ).
% insert_corr
thf(fact_517_vebt__insert_Osimps_I4_J,axiom,
! [V2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X3 @ X3 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary ) ) ).
% vebt_insert.simps(4)
thf(fact_518_semiring__norm_I75_J,axiom,
! [M2: num] :
~ ( ord_less @ num @ M2 @ one2 ) ).
% semiring_norm(75)
thf(fact_519_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% semiring_norm(68)
thf(fact_520_semiring__norm_I78_J,axiom,
! [M2: num,N: num] :
( ( ord_less @ num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M2 @ N ) ) ).
% semiring_norm(78)
thf(fact_521_semiring__norm_I71_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq @ num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less_eq @ num @ M2 @ N ) ) ).
% semiring_norm(71)
thf(fact_522_semiring__norm_I87_J,axiom,
! [M2: num,N: num] :
( ( ( bit0 @ M2 )
= ( bit0 @ N ) )
= ( M2 = N ) ) ).
% semiring_norm(87)
thf(fact_523_UnCI,axiom,
! [A: $tType,C3: A,B5: set @ A,A6: set @ A] :
( ( ~ ( member @ A @ C3 @ B5 )
=> ( member @ A @ C3 @ A6 ) )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) ) ) ).
% UnCI
thf(fact_524_Un__iff,axiom,
! [A: $tType,C3: A,A6: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( ( member @ A @ C3 @ A6 )
| ( member @ A @ C3 @ B5 ) ) ) ).
% Un_iff
thf(fact_525_semiring__norm_I83_J,axiom,
! [N: num] :
( one2
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_526_semiring__norm_I85_J,axiom,
! [M2: num] :
( ( bit0 @ M2 )
!= one2 ) ).
% semiring_norm(85)
thf(fact_527_Un__empty,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A6 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A6
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_528_Un__subset__iff,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) @ C4 )
= ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_529_Un__insert__right,axiom,
! [A: $tType,A6: set @ A,A3: A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ B5 ) )
= ( insert2 @ A @ A3 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) ) ) ).
% Un_insert_right
thf(fact_530_Un__insert__left,axiom,
! [A: $tType,A3: A,B5: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A3 @ B5 ) @ C4 )
= ( insert2 @ A @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Un_insert_left
thf(fact_531_semiring__norm_I6_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% semiring_norm(6)
thf(fact_532_psubsetD,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C3: A] :
( ( ord_less @ ( set @ A ) @ A6 @ B5 )
=> ( ( member @ A @ C3 @ A6 )
=> ( member @ A @ C3 @ B5 ) ) ) ).
% psubsetD
thf(fact_533_psubset__trans,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B5 )
=> ( ( ord_less @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% psubset_trans
thf(fact_534_UnE,axiom,
! [A: $tType,C3: A,A6: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
=> ( ~ ( member @ A @ C3 @ A6 )
=> ( member @ A @ C3 @ B5 ) ) ) ).
% UnE
thf(fact_535_UnI1,axiom,
! [A: $tType,C3: A,A6: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ A6 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) ) ) ).
% UnI1
thf(fact_536_UnI2,axiom,
! [A: $tType,C3: A,B5: set @ A,A6: set @ A] :
( ( member @ A @ C3 @ B5 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) ) ) ).
% UnI2
thf(fact_537_bex__Un,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
& ( P @ X4 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P @ X4 ) )
| ? [X4: A] :
( ( member @ A @ X4 @ B5 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_538_ball__Un,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( P @ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ B5 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_539_Un__assoc,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) @ C4 )
= ( sup_sup @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Un_assoc
thf(fact_540_Un__absorb,axiom,
! [A: $tType,A6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ A6 )
= A6 ) ).
% Un_absorb
thf(fact_541_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A7 ) ) ) ).
% Un_commute
thf(fact_542_Un__left__absorb,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ A6 @ B5 ) ) ).
% Un_left_absorb
thf(fact_543_Un__left__commute,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% Un_left_commute
thf(fact_544_Un__empty__right,axiom,
! [A: $tType,A6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
= A6 ) ).
% Un_empty_right
thf(fact_545_Un__empty__left,axiom,
! [A: $tType,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_546_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A7 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_547_subset__UnE,axiom,
! [A: $tType,C4: set @ A,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
=> ~ ! [A9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A9 @ A6 )
=> ! [B9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B9 @ B5 )
=> ( C4
!= ( sup_sup @ ( set @ A ) @ A9 @ B9 ) ) ) ) ) ).
% subset_UnE
thf(fact_548_Un__absorb2,axiom,
! [A: $tType,B5: set @ A,A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ( sup_sup @ ( set @ A ) @ A6 @ B5 )
= A6 ) ) ).
% Un_absorb2
thf(fact_549_Un__absorb1,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( sup_sup @ ( set @ A ) @ A6 @ B5 )
= B5 ) ) ).
% Un_absorb1
thf(fact_550_Un__upper2,axiom,
! [A: $tType,B5: set @ A,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) ) ).
% Un_upper2
thf(fact_551_Un__upper1,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) ) ).
% Un_upper1
thf(fact_552_Un__least,axiom,
! [A: $tType,A6: set @ A,C4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) @ C4 ) ) ) ).
% Un_least
thf(fact_553_Un__mono,axiom,
! [A: $tType,A6: set @ A,C4: set @ A,B5: set @ A,D4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) @ ( sup_sup @ ( set @ A ) @ C4 @ D4 ) ) ) ) ).
% Un_mono
thf(fact_554_singleton__Un__iff,axiom,
! [A: $tType,X3: A,A6: set @ A,B5: set @ A] :
( ( ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( ( ( A6
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A6
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A6
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_555_Un__singleton__iff,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,X3: A] :
( ( ( sup_sup @ ( set @ A ) @ A6 @ B5 )
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A6
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A6
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A6
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_556_insert__is__Un,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A8: A] : ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A8 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_557_delete__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X3 ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert2 @ nat @ X3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% delete_correct
thf(fact_558_case4_I11_J,axiom,
( ( mi != ma )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( ( ( vEBT_VEBT_high @ ma @ na )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList2 @ I2 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
& ! [X: nat] :
( ( ( ( vEBT_VEBT_high @ X @ na )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList2 @ I2 ) @ ( vEBT_VEBT_low @ X @ na ) ) )
=> ( ( ord_less @ nat @ mi @ X )
& ( ord_less_eq @ nat @ X @ ma ) ) ) ) ) ) ).
% case4(11)
thf(fact_559_enat__ord__number_I1_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( numeral_numeral @ extended_enat @ N ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_560_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_561_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A3 @ B2 ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_562_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_563_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A,B2: A] :
( ( ( sup_sup @ A @ A3 @ B2 )
= ( bot_bot @ A ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_564_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X3: A,Y: A] :
( ( ( sup_sup @ A @ X3 @ Y )
= ( bot_bot @ A ) )
= ( ( X3
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_565_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X3: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X3 @ Y ) )
= ( ( X3
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_566_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ ( bot_bot @ A ) )
= X3 ) ) ).
% sup_bot_right
thf(fact_567_bit__split__inv,axiom,
! [X3: nat,D3: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X3 @ D3 ) @ ( vEBT_VEBT_low @ X3 @ D3 ) @ D3 )
= X3 ) ).
% bit_split_inv
thf(fact_568_Diff__idemp,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A6 @ B5 ) ) ).
% Diff_idemp
thf(fact_569_Diff__iff,axiom,
! [A: $tType,C3: A,A6: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) )
= ( ( member @ A @ C3 @ A6 )
& ~ ( member @ A @ C3 @ B5 ) ) ) ).
% Diff_iff
thf(fact_570_DiffI,axiom,
! [A: $tType,C3: A,A6: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ A6 )
=> ( ~ ( member @ A @ C3 @ B5 )
=> ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) ) ) ) ).
% DiffI
thf(fact_571_high__bound__aux,axiom,
! [Ma: nat,N: nat,M2: nat] :
( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M2 ) ) )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ).
% high_bound_aux
thf(fact_572_delete__correct_H,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X3 ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( insert2 @ nat @ X3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% delete_correct'
thf(fact_573_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% add_diff_cancel
thf(fact_574_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% diff_add_cancel
thf(fact_575_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B2 ) )
= ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_576_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ A3 )
= B2 ) ) ).
% add_diff_cancel_left'
thf(fact_577_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_578_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% add_diff_cancel_right'
thf(fact_579_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X3 @ Z2 )
& ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% le_sup_iff
thf(fact_580_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A3 )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% sup.bounded_iff
thf(fact_581_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X3 )
= X3 ) ) ).
% sup_bot_left
thf(fact_582_Diff__empty,axiom,
! [A: $tType,A6: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
= A6 ) ).
% Diff_empty
thf(fact_583_empty__Diff,axiom,
! [A: $tType,A6: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A6 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_584_Diff__cancel,axiom,
! [A: $tType,A6: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ A6 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_585_Diff__insert0,axiom,
! [A: $tType,X3: A,A6: set @ A,B5: set @ A] :
( ~ ( member @ A @ X3 @ A6 )
=> ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ A6 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_586_insert__Diff1,axiom,
! [A: $tType,X3: A,B5: set @ A,A6: set @ A] :
( ( member @ A @ X3 @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X3 @ A6 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A6 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_587_Un__Diff__cancel2,axiom,
! [A: $tType,B5: set @ A,A6: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B5 @ A6 ) @ A6 )
= ( sup_sup @ ( set @ A ) @ B5 @ A6 ) ) ).
% Un_Diff_cancel2
thf(fact_588_Un__Diff__cancel,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ ( minus_minus @ ( set @ A ) @ B5 @ A6 ) )
= ( sup_sup @ ( set @ A ) @ A6 @ B5 ) ) ).
% Un_Diff_cancel
thf(fact_589_Diff__eq__empty__iff,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A6 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A6 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_590_insert__Diff__single,axiom,
! [A: $tType,A3: A,A6: set @ A] :
( ( insert2 @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert2 @ A @ A3 @ A6 ) ) ).
% insert_Diff_single
thf(fact_591_enat__ord__number_I2_J,axiom,
! [M2: num,N: num] :
( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( numeral_numeral @ extended_enat @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_592_acd,axiom,
( ( mi != ma )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( ( ( vEBT_VEBT_high @ ma @ na )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
& ! [X: nat] :
( ( ( ( vEBT_VEBT_high @ X @ na )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ X @ na ) ) )
=> ( ( ord_less @ nat @ mi @ X )
& ( ord_less_eq @ nat @ X @ ma ) ) ) ) ) ) ).
% acd
thf(fact_593_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C3 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% diff_right_commute
thf(fact_594_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( A3 = B2 )
= ( C3 = D3 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_595_DiffD2,axiom,
! [A: $tType,C3: A,A6: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) )
=> ~ ( member @ A @ C3 @ B5 ) ) ).
% DiffD2
thf(fact_596_DiffD1,axiom,
! [A: $tType,C3: A,A6: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) )
=> ( member @ A @ C3 @ A6 ) ) ).
% DiffD1
thf(fact_597_DiffE,axiom,
! [A: $tType,C3: A,A6: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) )
=> ~ ( ( member @ A @ C3 @ A6 )
=> ( member @ A @ C3 @ B5 ) ) ) ).
% DiffE
thf(fact_598_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ D3 @ C3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C3 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_599_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C3 @ A3 ) @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_600_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C3 ) @ ( minus_minus @ A @ B2 @ C3 ) ) ) ) ).
% diff_right_mono
thf(fact_601_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
= ( ord_less_eq @ A @ C3 @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_602_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C3 ) @ ( minus_minus @ A @ B2 @ C3 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_603_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less @ A @ ( minus_minus @ A @ C3 @ A3 ) @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_604_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
= ( ord_less @ A @ C3 @ D3 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_605_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D3: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ D3 @ C3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C3 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_606_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A6: A,K2: A,A3: A,B2: A] :
( ( A6
= ( plus_plus @ A @ K2 @ A3 ) )
=> ( ( minus_minus @ A @ A6 @ B2 )
= ( plus_plus @ A @ K2 @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_607_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= C3 )
= ( A3
= ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_608_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( A3
= ( minus_minus @ A @ C3 @ B2 ) )
= ( ( plus_plus @ A @ A3 @ B2 )
= C3 ) ) ) ).
% eq_diff_eq
thf(fact_609_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% add_diff_eq
thf(fact_610_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( minus_minus @ A @ A3 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_611_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C3 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_612_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C3 ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_613_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ( plus_plus @ A @ C3 @ B2 )
= A3 )
=> ( C3
= ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_614_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C3 )
= ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% diff_diff_eq
thf(fact_615_Diff__mono,axiom,
! [A: $tType,A6: set @ A,C4: set @ A,D4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ D4 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D4 ) ) ) ) ).
% Diff_mono
thf(fact_616_Diff__subset,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) @ A6 ) ).
% Diff_subset
thf(fact_617_double__diff,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ( minus_minus @ ( set @ A ) @ B5 @ ( minus_minus @ ( set @ A ) @ C4 @ A6 ) )
= A6 ) ) ) ).
% double_diff
thf(fact_618_insert__Diff__if,axiom,
! [A: $tType,X3: A,B5: set @ A,A6: set @ A] :
( ( ( member @ A @ X3 @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X3 @ A6 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A6 @ B5 ) ) )
& ( ~ ( member @ A @ X3 @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X3 @ A6 ) @ B5 )
= ( insert2 @ A @ X3 @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_619_Un__Diff,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) @ C4 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ C4 ) @ ( minus_minus @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Un_Diff
thf(fact_620_psubset__imp__ex__mem,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B5 )
=> ? [B4: A] : ( member @ A @ B4 @ ( minus_minus @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_621_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C3 )
= ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% diff_le_eq
thf(fact_622_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( minus_minus @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% le_diff_eq
thf(fact_623_diff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% diff_add
thf(fact_624_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C3 ) @ A3 ) ) ) ) ).
% le_add_diff
thf(fact_625_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_626_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ C3 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C3 @ B2 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_627_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C3 @ B2 ) @ A3 )
= ( plus_plus @ A @ C3 @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_628_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C3 )
= ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C3 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_629_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C3 ) @ A3 )
= ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_630_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ C3 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C3 @ A3 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_631_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ A3 ) )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_632_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( minus_minus @ A @ B2 @ A3 )
= C3 )
= ( B2
= ( plus_plus @ A @ C3 @ A3 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_633_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( minus_minus @ A @ C3 @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% less_diff_eq
thf(fact_634_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C3 )
= ( ord_less @ A @ A3 @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% diff_less_eq
thf(fact_635_Diff__insert,axiom,
! [A: $tType,A6: set @ A,A3: A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_636_insert__Diff,axiom,
! [A: $tType,A3: A,A6: set @ A] :
( ( member @ A @ A3 @ A6 )
=> ( ( insert2 @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A6 ) ) ).
% insert_Diff
thf(fact_637_Diff__insert2,axiom,
! [A: $tType,A6: set @ A,A3: A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_638_Diff__insert__absorb,axiom,
! [A: $tType,X3: A,A6: set @ A] :
( ~ ( member @ A @ X3 @ A6 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X3 @ A6 ) @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= A6 ) ) ).
% Diff_insert_absorb
thf(fact_639_subset__Diff__insert,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,X3: A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( minus_minus @ ( set @ A ) @ B5 @ ( insert2 @ A @ X3 @ C4 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( minus_minus @ ( set @ A ) @ B5 @ C4 ) )
& ~ ( member @ A @ X3 @ A6 ) ) ) ).
% subset_Diff_insert
thf(fact_640_Diff__subset__conv,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) @ C4 )
= ( ord_less_eq @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Diff_subset_conv
thf(fact_641_Diff__partition,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( sup_sup @ ( set @ A ) @ A6 @ ( minus_minus @ ( set @ A ) @ B5 @ A6 ) )
= B5 ) ) ).
% Diff_partition
thf(fact_642_subset__insert__iff,axiom,
! [A: $tType,A6: set @ A,X3: A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ B5 ) )
= ( ( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) )
& ( ~ ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_643_Diff__single__insert,axiom,
! [A: $tType,A6: set @ A,X3: A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_644_power2__commute,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X3: A,Y: A] :
( ( power_power @ A @ ( minus_minus @ A @ X3 @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ ( minus_minus @ A @ Y @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_commute
thf(fact_645_psubset__insert__iff,axiom,
! [A: $tType,A6: set @ A,X3: A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ B5 ) )
= ( ( ( member @ A @ X3 @ B5 )
=> ( ord_less @ ( set @ A ) @ A6 @ B5 ) )
& ( ~ ( member @ A @ X3 @ B5 )
=> ( ( ( member @ A @ X3 @ A6 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) )
& ( ~ ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_646_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X3: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X3 @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_647_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_648_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A,X3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq @ A @ A3 @ X3 )
=> ~ ( ord_less_eq @ A @ B2 @ X3 ) ) ) ) ).
% le_supE
thf(fact_649_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,X3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ X3 )
=> ( ( ord_less_eq @ A @ B2 @ X3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X3 ) ) ) ) ).
% le_supI
thf(fact_650_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y ) ) ) ).
% sup_ge1
thf(fact_651_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X3: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X3 @ Y ) ) ) ).
% sup_ge2
thf(fact_652_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X3 @ A3 )
=> ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_653_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ X3 @ B2 )
=> ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_654_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A3: A,D3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ( ord_less_eq @ A @ D3 @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C3 @ D3 ) @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_655_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,C3: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ ( sup_sup @ A @ C3 @ D3 ) ) ) ) ) ).
% sup_mono
thf(fact_656_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X3: A,Z2: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X3 ) ) ) ) ).
% sup_least
thf(fact_657_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y3: A] :
( ( sup_sup @ A @ X4 @ Y3 )
= Y3 ) ) ) ) ).
% le_iff_sup
thf(fact_658_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3
= ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.orderE
thf(fact_659_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( sup_sup @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% sup.orderI
thf(fact_660_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F3: A > A > A,X3: A,Y: A] :
( ! [X5: A,Y4: A] : ( ord_less_eq @ A @ X5 @ ( F3 @ X5 @ Y4 ) )
=> ( ! [X5: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ ( F3 @ X5 @ Y4 ) )
=> ( ! [X5: A,Y4: A,Z3: A] :
( ( ord_less_eq @ A @ Y4 @ X5 )
=> ( ( ord_less_eq @ A @ Z3 @ X5 )
=> ( ord_less_eq @ A @ ( F3 @ Y4 @ Z3 ) @ X5 ) ) )
=> ( ( sup_sup @ A @ X3 @ Y )
= ( F3 @ X3 @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_661_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb1
thf(fact_662_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb2
thf(fact_663_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( sup_sup @ A @ X3 @ Y )
= X3 ) ) ) ).
% sup_absorb1
thf(fact_664_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( sup_sup @ A @ X3 @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_665_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A3 )
=> ~ ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% sup.boundedE
thf(fact_666_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A3 ) ) ) ) ).
% sup.boundedI
thf(fact_667_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B8: A,A8: A] :
( A8
= ( sup_sup @ A @ A8 @ B8 ) ) ) ) ) ).
% sup.order_iff
thf(fact_668_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.cobounded1
thf(fact_669_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.cobounded2
thf(fact_670_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B8: A,A8: A] :
( ( sup_sup @ A @ A8 @ B8 )
= A8 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_671_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] :
( ( sup_sup @ A @ A8 @ B8 )
= B8 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_672_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_673_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_674_invar__vebt_Ointros_I4_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2 = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M2 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_675_invar__vebt_Ointros_I5_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M2 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_676_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N3: nat,TreeList4: list @ vEBT_VEBT,X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ X4 @ N3 ) ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) ) ) ).
% in_children_def
thf(fact_677_both__member__options__from__chilf__to__complete__tree,axiom,
! [X3: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_678_member__inv,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
& ( ( X3 = Mi )
| ( X3 = Ma )
| ( ( ord_less @ nat @ X3 @ Ma )
& ( ord_less @ nat @ Mi @ X3 )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% member_inv
thf(fact_679_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
| ( X3 = Mi )
| ( X3 = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_680_pred__list__to__short,axiom,
! [Deg: nat,X3: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X3 @ Ma )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( none @ nat ) ) ) ) ) ).
% pred_list_to_short
thf(fact_681_succ__list__to__short,axiom,
! [Deg: nat,Mi: nat,X3: nat,TreeList: list @ vEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X3 )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( none @ nat ) ) ) ) ) ).
% succ_list_to_short
thf(fact_682_both__member__options__ding,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X3 ) ) ) ) ).
% both_member_options_ding
thf(fact_683_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
= A3 ) ) ) ).
% le_add_diff_inverse
thf(fact_684_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ) ).
% le_add_diff_inverse2
thf(fact_685_low__inv,axiom,
! [X3: nat,N: nat,Y: nat] :
( ( ord_less @ nat @ X3 @ ( 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 ) ) @ X3 ) @ N )
= X3 ) ) ).
% low_inv
thf(fact_686_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_687_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus @ nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_688_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_689_diff__diff__left,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K2 )
= ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_690_power__minus__is__div,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq @ nat @ B2 @ A3 )
=> ( ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% power_minus_is_div
thf(fact_691_pow__sum,axiom,
! [A3: nat,B2: nat] :
( ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ A3 @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) ) ).
% pow_sum
thf(fact_692_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X4: nat,N3: nat] : ( divide_divide @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% high_def
thf(fact_693_high__inv,axiom,
! [X3: nat,N: nat,Y: nat] :
( ( ord_less @ nat @ X3 @ ( 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 ) ) @ X3 ) @ N )
= Y ) ) ).
% high_inv
thf(fact_694_numeral__times__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [M2: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M2 @ N ) ) ) ) ).
% numeral_times_numeral
thf(fact_695_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [V2: num,W: num,Z2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Z2 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) @ Z2 ) ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_696_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.right_neutral
thf(fact_697_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% mult_1
thf(fact_698_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_699_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( minus_minus @ nat @ I @ ( minus_minus @ nat @ J @ K2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_700_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K2 ) @ I )
= ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_701_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_702_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_703_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M2 @ N ) )
= ( ( M2
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_704_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times @ nat @ M2 @ N )
= ( one_one @ nat ) )
= ( ( M2
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_705_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A3: A,B2: A,V2: num] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_706_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V2: num,B2: A,C3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C3 ) ) ) ) ).
% distrib_left_numeral
thf(fact_707_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [A3: A,B2: A,V2: num] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_708_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [V2: num,B2: A,C3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C3 ) ) ) ) ).
% right_diff_distrib_numeral
thf(fact_709_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J @ K2 ) ) @ I )
= ( minus_minus @ nat @ ( suc @ J ) @ ( plus_plus @ nat @ K2 @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_710_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( minus_minus @ nat @ I @ ( suc @ ( minus_minus @ nat @ J @ K2 ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_711_mult__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( times_times @ nat @ M2 @ ( suc @ N ) )
= ( plus_plus @ nat @ M2 @ ( times_times @ nat @ M2 @ N ) ) ) ).
% mult_Suc_right
thf(fact_712_mintlistlength,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 )
=> ( ( Mi != Ma )
=> ( ( ord_less @ nat @ Mi @ Ma )
& ? [M: nat] :
( ( ( some @ nat @ M )
= ( vEBT_vebt_mint @ Summary ) )
& ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% mintlistlength
thf(fact_713_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) @ B2 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_714_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) @ A3 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_715_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) @ A3 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_716_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) @ B2 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_717_power__add__numeral2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M2: num,N: num,B2: A] :
( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M2 ) ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M2 @ N ) ) ) @ B2 ) ) ) ).
% power_add_numeral2
thf(fact_718_power__add__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M2: num,N: num] :
( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M2 ) ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% power_add_numeral
thf(fact_719_diff__mult__distrib2,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( times_times @ nat @ K2 @ ( minus_minus @ nat @ M2 @ N ) )
= ( minus_minus @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_720_diff__mult__distrib,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ M2 @ N ) @ K2 )
= ( minus_minus @ nat @ ( times_times @ nat @ M2 @ K2 ) @ ( times_times @ nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_721_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_722_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% mult.assoc
thf(fact_723_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A8: A,B8: A] : ( times_times @ A @ B8 @ A8 ) ) ) ) ).
% mult.commute
thf(fact_724_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% mult.left_commute
thf(fact_725_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,E3: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ E3 ) @ C3 ) ) ) ).
% combine_common_factor
thf(fact_726_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% distrib_right
thf(fact_727_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ).
% distrib_left
thf(fact_728_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_729_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_730_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_731_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( minus_minus @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_732_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_733_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_734_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( 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 ) @ M2 ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_735_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_736_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_737_power__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ A3 @ B2 ) @ N )
= ( divide_divide @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% power_divide
thf(fact_738_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus @ nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_739_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_740_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_741_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( ord_less @ nat @ M2 @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_742_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less @ nat @ J @ K2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_743_power__commuting__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X3: A,Y: A,N: nat] :
( ( ( times_times @ A @ X3 @ Y )
= ( times_times @ A @ Y @ X3 ) )
=> ( ( times_times @ A @ ( power_power @ A @ X3 @ N ) @ Y )
= ( times_times @ A @ Y @ ( power_power @ A @ X3 @ N ) ) ) ) ) ).
% power_commuting_commutes
thf(fact_744_power__mult__distrib,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( power_power @ A @ ( times_times @ A @ A3 @ B2 ) @ N )
= ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% power_mult_distrib
thf(fact_745_power__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 )
= ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_commutes
thf(fact_746_Suc__mult__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( times_times @ nat @ ( suc @ K2 ) @ M2 )
= ( times_times @ nat @ ( suc @ K2 ) @ N ) )
= ( M2 = N ) ) ).
% Suc_mult_cancel1
thf(fact_747_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_748_le__diff__iff_H,axiom,
! [A3: nat,C3: nat,B2: nat] :
( ( ord_less_eq @ nat @ A3 @ C3 )
=> ( ( ord_less_eq @ nat @ B2 @ C3 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C3 @ A3 ) @ ( minus_minus @ nat @ C3 @ B2 ) )
= ( ord_less_eq @ nat @ B2 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_749_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_750_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_751_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M2 )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
= ( minus_minus @ nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_752_le__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M2 )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_753_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M2 )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( ( minus_minus @ nat @ M2 @ K2 )
= ( minus_minus @ nat @ N @ K2 ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_754_power__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( power_power @ A @ A3 @ ( times_times @ nat @ M2 @ N ) )
= ( power_power @ A @ ( power_power @ A @ A3 @ M2 ) @ N ) ) ) ).
% power_mult
thf(fact_755_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_756_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_757_diff__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) )
= ( minus_minus @ nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_758_Nat_Odiff__cancel,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ K2 @ M2 ) @ ( plus_plus @ nat @ K2 @ N ) )
= ( minus_minus @ nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_759_mult__le__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ K2 @ I ) @ ( times_times @ nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_760_mult__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K2 ) @ ( times_times @ nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_761_mult__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K2 @ L )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K2 ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_762_le__square,axiom,
! [M2: nat] : ( ord_less_eq @ nat @ M2 @ ( times_times @ nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_763_le__cube,axiom,
! [M2: nat] : ( ord_less_eq @ nat @ M2 @ ( times_times @ nat @ M2 @ ( times_times @ nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_764_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K2: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K2 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K2 ) ) ).
% left_add_mult_distrib
thf(fact_765_add__mult__distrib2,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( times_times @ nat @ K2 @ ( plus_plus @ nat @ M2 @ N ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_766_add__mult__distrib,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M2 @ N ) @ K2 )
= ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K2 ) @ ( times_times @ nat @ N @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_767_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_768_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_769_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E3 ) @ C3 )
= D3 ) ) ) ).
% eq_add_iff1
thf(fact_770_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( C3
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E3 ) @ D3 ) ) ) ) ).
% eq_add_iff2
thf(fact_771_square__diff__square__factored,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [X3: A,Y: A] :
( ( minus_minus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) )
= ( times_times @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( minus_minus @ A @ X3 @ Y ) ) ) ) ).
% square_diff_square_factored
thf(fact_772_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_773_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_774_ordered__ring__class_Ole__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less_eq @ A @ C3 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E3 ) @ D3 ) ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_775_ordered__ring__class_Ole__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E3 ) @ C3 ) @ D3 ) ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_776_divide__numeral__1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
= A3 ) ) ).
% divide_numeral_1
thf(fact_777_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less @ A @ C3 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E3 ) @ D3 ) ) ) ) ).
% less_add_iff2
thf(fact_778_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E3 ) @ C3 ) @ D3 ) ) ) ).
% less_add_iff1
thf(fact_779_square__diff__one__factored,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: A] :
( ( minus_minus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( one_one @ A ) )
= ( times_times @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) @ ( minus_minus @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% square_diff_one_factored
thf(fact_780_power__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_one_over
thf(fact_781_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_782_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_783_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ N @ M2 )
=> ( ( suc @ ( minus_minus @ nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus @ nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_784_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_785_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X3: A,Y: A,N: nat] :
( ( ( times_times @ A @ X3 @ Y )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X3 @ N ) @ ( power_power @ A @ Y @ N ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_786_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( minus_minus @ nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus @ nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_787_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_Suc
thf(fact_788_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 ) ) ) ).
% power_Suc2
thf(fact_789_less__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M2 )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_790_diff__less__mono,axiom,
! [A3: nat,B2: nat,C3: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( ( ord_less_eq @ nat @ C3 @ A3 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A3 @ C3 ) @ ( minus_minus @ nat @ B2 @ C3 ) ) ) ) ).
% diff_less_mono
thf(fact_791_less__diff__conv,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K2 ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ I @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_792_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less @ nat @ M2 @ N )
=> ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_793_Suc__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K2 ) @ M2 ) @ ( times_times @ nat @ ( suc @ K2 ) @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_794_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( minus_minus @ nat @ J @ I )
= K2 )
= ( J
= ( plus_plus @ nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_795_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K2 )
= ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_796_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K2 )
= ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_797_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ K2 ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_798_le__diff__conv,axiom,
! [J: nat,K2: nat,I: nat] :
( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K2 ) @ I )
= ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_799_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( power_power @ A @ A3 @ ( plus_plus @ nat @ M2 @ N ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_add
thf(fact_800_Suc__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K2 ) @ M2 ) @ ( times_times @ nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_801_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus @ nat @ M2 @ ( suc @ N ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_802_mult__Suc,axiom,
! [M2: nat,N: nat] :
( ( times_times @ nat @ ( suc @ M2 ) @ N )
= ( plus_plus @ nat @ N @ ( times_times @ nat @ M2 @ N ) ) ) ).
% mult_Suc
thf(fact_803_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ ( power_power @ A @ A3 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_804_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_805_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_806_less__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K2 ) @ I )
= ( ord_less @ nat @ J @ ( plus_plus @ nat @ I @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_807_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ B2 ) ) ) ).
% left_add_twice
thf(fact_808_mult__2__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z2: A] :
( ( times_times @ A @ Z2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ Z2 @ Z2 ) ) ) ).
% mult_2_right
thf(fact_809_mult__2,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 )
= ( plus_plus @ A @ Z2 @ Z2 ) ) ) ).
% mult_2
thf(fact_810_power2__eq__square,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ A3 @ A3 ) ) ) ).
% power2_eq_square
thf(fact_811_power4__eq__xxxx,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X3: A] :
( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( times_times @ A @ ( times_times @ A @ X3 @ X3 ) @ X3 ) @ X3 ) ) ) ).
% power4_eq_xxxx
thf(fact_812_power__even__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ A @ ( power_power @ A @ A3 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power_even_eq
thf(fact_813_diff__le__diff__pow,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ N ) @ ( minus_minus @ nat @ ( power_power @ nat @ K2 @ M2 ) @ ( power_power @ nat @ K2 @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_814_power2__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X3: A,Y: A] :
( ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) @ Y ) ) ) ) ).
% power2_sum
thf(fact_815_power2__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X3: A,Y: A] :
( ( power_power @ A @ ( minus_minus @ A @ X3 @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) @ Y ) ) ) ) ).
% power2_diff
thf(fact_816_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K2: A,N: A,J: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K2 ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K2 ) )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K2 ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K2 ) )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K2 ) @ J ) ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_817_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K2: A,N: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K2 ) @ N )
=> ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N @ K2 ) ) ) ) ).
% add_le_imp_le_diff
thf(fact_818_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_819_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_820_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A] :
( ~ ( ord_less @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
= A3 ) ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_821_add__self__div__2,axiom,
! [M2: nat] :
( ( divide_divide @ nat @ ( plus_plus @ nat @ M2 @ M2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= M2 ) ).
% add_self_div_2
thf(fact_822_div2__Suc__Suc,axiom,
! [M2: nat] :
( ( divide_divide @ nat @ ( suc @ ( suc @ M2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% div2_Suc_Suc
thf(fact_823_nested__mint,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( N
= ( suc @ ( suc @ Va ) ) )
=> ( ~ ( ord_less @ nat @ Ma @ Mi )
=> ( ( Ma != Mi )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide @ nat @ Va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ).
% nested_mint
thf(fact_824_sum__squares__bound,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) @ Y ) @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_squares_bound
thf(fact_825_mul__def,axiom,
( vEBT_VEBT_mul
= ( vEBT_V2048590022279873568_shift @ nat @ ( times_times @ nat ) ) ) ).
% mul_def
thf(fact_826_summaxma,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 )
=> ( ( the2 @ nat @ ( vEBT_vebt_maxt @ Summary ) )
= ( vEBT_VEBT_high @ Ma @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% summaxma
thf(fact_827_div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ).
% div_exp_eq
thf(fact_828_field__less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ A @ X3 @ ( divide_divide @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% field_less_half_sum
thf(fact_829_div__nat__eqI,axiom,
! [N: nat,Q3: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q3 ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ ( times_times @ nat @ N @ ( suc @ Q3 ) ) )
=> ( ( divide_divide @ nat @ M2 @ N )
= Q3 ) ) ) ).
% div_nat_eqI
thf(fact_830_Suc__double__not__eq__double,axiom,
! [M2: nat,N: nat] :
( ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
!= ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_831_mul__shift,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ( times_times @ nat @ X3 @ Y )
= Z2 )
= ( ( vEBT_VEBT_mul @ ( some @ nat @ X3 ) @ ( some @ nat @ Y ) )
= ( some @ nat @ Z2 ) ) ) ).
% mul_shift
thf(fact_832_semiring__norm_I13_J,axiom,
! [M2: num,N: num] :
( ( times_times @ num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times @ num @ M2 @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_833_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times @ num @ one2 @ N )
= N ) ).
% semiring_norm(12)
thf(fact_834_semiring__norm_I11_J,axiom,
! [M2: num] :
( ( times_times @ num @ M2 @ one2 )
= M2 ) ).
% semiring_norm(11)
thf(fact_835_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the2 @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_836_num__double,axiom,
! [N: num] :
( ( times_times @ num @ ( bit0 @ one2 ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_837_power__mult__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M2: num,N: num] :
( ( power_power @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M2 ) ) @ ( numeral_numeral @ nat @ N ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( times_times @ num @ M2 @ N ) ) ) ) ) ).
% power_mult_numeral
thf(fact_838_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K2 )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_839_div__mult2__numeral__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,K2: num,L: num] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ K2 ) ) @ ( numeral_numeral @ A @ L ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( times_times @ num @ K2 @ L ) ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_840_option_Osel,axiom,
! [A: $tType,X2: A] :
( ( the2 @ A @ ( some @ A @ X2 ) )
= X2 ) ).
% option.sel
thf(fact_841_option_Oexpand,axiom,
! [A: $tType,Option: option @ A,Option2: option @ A] :
( ( ( Option
= ( none @ A ) )
= ( Option2
= ( none @ A ) ) )
=> ( ( ( Option
!= ( none @ A ) )
=> ( ( Option2
!= ( none @ A ) )
=> ( ( the2 @ A @ Option )
= ( the2 @ A @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_842_L2__set__mult__ineq__lemma,axiom,
! [A3: real,C3: real,B2: real,D3: real] : ( ord_less_eq @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( times_times @ real @ A3 @ C3 ) ) @ ( times_times @ real @ B2 @ D3 ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( power_power @ real @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ real @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ C3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_843_four__x__squared,axiom,
! [X3: real] :
( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% four_x_squared
thf(fact_844_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_845_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_846_add__diff__add,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,C3: A,B2: A,D3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) )
= ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( minus_minus @ A @ C3 @ D3 ) ) ) ) ).
% add_diff_add
thf(fact_847_div__le__mono,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ M2 @ K2 ) @ ( divide_divide @ nat @ N @ K2 ) ) ) ).
% div_le_mono
thf(fact_848_div__le__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M2 @ N ) @ M2 ) ).
% div_le_dividend
thf(fact_849_mult__diff__mult,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [X3: A,Y: A,A3: A,B2: A] :
( ( minus_minus @ A @ ( times_times @ A @ X3 @ Y ) @ ( times_times @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ X3 @ ( minus_minus @ A @ Y @ B2 ) ) @ ( times_times @ A @ ( minus_minus @ A @ X3 @ A3 ) @ B2 ) ) ) ) ).
% mult_diff_mult
thf(fact_850_Suc__div__le__mono,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M2 @ N ) @ ( divide_divide @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_851_times__div__less__eq__dividend,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M2 @ N ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_852_div__times__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M2 @ N ) @ N ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_853_numeral__Bit0__div__2,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% numeral_Bit0_div_2
thf(fact_854_field__sum__of__halves,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ A @ X3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= X3 ) ) ).
% field_sum_of_halves
thf(fact_855_double__not__eq__Suc__double,axiom,
! [M2: nat,N: nat] :
( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 )
!= ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_856_del__x__mi__lets__in__not__minNull,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( X3 = Mi )
& ( ord_less @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H @ Newnode ) )
=> ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi_lets_in_not_minNull
thf(fact_857_del__x__not__mi__newnode__not__nil,axiom,
! [Mi: nat,X3: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Newlist: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H ) @ L ) )
=> ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X3 = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_not_mi_newnode_not_nil
thf(fact_858_real__average__minus__first,axiom,
! [A3: real,B2: real] :
( ( minus_minus @ real @ ( divide_divide @ real @ ( plus_plus @ real @ A3 @ B2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ A3 )
= ( divide_divide @ real @ ( minus_minus @ real @ B2 @ A3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% real_average_minus_first
thf(fact_859_real__average__minus__second,axiom,
! [B2: real,A3: real] :
( ( minus_minus @ real @ ( divide_divide @ real @ ( plus_plus @ real @ B2 @ A3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ A3 )
= ( divide_divide @ real @ ( minus_minus @ real @ B2 @ A3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% real_average_minus_second
thf(fact_860_vebt__pred_Osimps_I4_J,axiom,
! [Uy: nat,Uz: list @ vEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz @ Va ) @ Vb )
= ( none @ nat ) ) ).
% vebt_pred.simps(4)
thf(fact_861_vebt__succ_Osimps_I3_J,axiom,
! [Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz ) @ Va )
= ( none @ nat ) ) ).
% vebt_succ.simps(3)
thf(fact_862_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R2: A,Q3: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R2 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ ( 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 @ Q3 @ R2 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ R2 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_863_pred__lesseq__max,axiom,
! [Deg: nat,X3: nat,Ma: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X3 @ Ma )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X3 ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% pred_lesseq_max
thf(fact_864_pred__less__length__list,axiom,
! [Deg: nat,X3: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X3 @ Ma )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X3 ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% pred_less_length_list
thf(fact_865_succ__less__length__list,axiom,
! [Deg: nat,Mi: nat,X3: nat,TreeList: list @ vEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X3 )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% succ_less_length_list
thf(fact_866_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% set_vebt'_def
thf(fact_867_zdiv__numeral__Bit0,axiom,
! [V2: num,W: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit0 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_868_succ__empty,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X3 )
= ( none @ nat ) )
= ( ( collect @ nat
@ ^ [Y3: nat] :
( ( vEBT_vebt_member @ T2 @ Y3 )
& ( ord_less @ nat @ X3 @ Y3 ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% succ_empty
thf(fact_869_pred__empty,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X3 )
= ( none @ nat ) )
= ( ( collect @ nat
@ ^ [Y3: nat] :
( ( vEBT_vebt_member @ T2 @ Y3 )
& ( ord_less @ nat @ Y3 @ X3 ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% pred_empty
thf(fact_870_singleton__conv2,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ( ^ [Y5: A,Z: A] : Y5 = Z
@ A3 ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_871_singleton__conv,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ^ [X4: A] : X4 = A3 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_872_del__x__not__mia,axiom,
! [Mi: nat,X3: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H ) @ L ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X3 = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) )
= ( none @ nat ) )
@ Mi
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H ) @ L ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H ) @ L ) )
@ ( vEBT_vebt_delete @ Summary @ H ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X3 = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H ) @ L ) ) @ H ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).
% del_x_not_mia
thf(fact_873_del__x__not__mi__new__node__nil,axiom,
! [Mi: nat,X3: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H ) @ L ) )
=> ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Sn
= ( vEBT_vebt_delete @ Summary @ H ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X3 = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ Sn )
= ( none @ nat ) )
@ Mi
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_not_mi_new_node_nil
thf(fact_874_del__x__not__mi,axiom,
! [Mi: nat,X3: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Newlist: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X3 = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) )
= ( none @ nat ) )
@ Mi
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ ( vEBT_vebt_delete @ Summary @ H ) ) ) )
& ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X3 = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_not_mi
thf(fact_875_del__in__range,axiom,
! [Mi: nat,X3: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less_eq @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X3 = Mi ) @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
@ ( if @ nat
@ ( ( ( X3 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X3 != Mi )
=> ( X3 = Ma ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X3 = Mi ) @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X3 = Mi ) @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
@ ( if @ nat
@ ( ( ( X3 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X3 != Mi )
=> ( X3 = Ma ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).
% del_in_range
thf(fact_876_del__x__mi,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L: nat] :
( ( ( X3 = Mi )
& ( ord_less @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H ) @ L ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Xn
@ ( if @ nat @ ( Xn = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) )
= ( none @ nat ) )
@ Xn
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H ) @ L ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H ) @ L ) )
@ ( vEBT_vebt_delete @ Summary @ H ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H ) @ L ) ) @ H ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi
thf(fact_877_del__x__mi__lets__in,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( X3 = Mi )
& ( ord_less @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H @ Newnode ) )
=> ( ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Xn
@ ( if @ nat @ ( Xn = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) )
= ( none @ nat ) )
@ Xn
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ ( vEBT_vebt_delete @ Summary @ H ) ) ) )
& ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi_lets_in
thf(fact_878_del__x__mi__lets__in__minNull,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT,Sn: vEBT_VEBT] :
( ( ( X3 = Mi )
& ( ord_less @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H @ Newnode ) )
=> ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Sn
= ( vEBT_vebt_delete @ Summary @ H ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Xn
@ ( if @ nat @ ( Xn = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ Sn )
= ( none @ nat ) )
@ Xn
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi_lets_in_minNull
thf(fact_879_del__x__mia,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X3 = Mi )
& ( ord_less @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( if @ nat
@ ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( if @ nat
@ ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).
% del_x_mia
thf(fact_880_succ__greatereq__min,axiom,
! [Deg: nat,Mi: nat,X3: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X3 )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% succ_greatereq_min
thf(fact_881_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_882_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A7 )
& ~ ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_883_real__arch__pow,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ? [N2: nat] : ( ord_less @ real @ Y @ ( power_power @ real @ X3 @ N2 ) ) ) ).
% real_arch_pow
thf(fact_884_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R )
@ ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ S3 ) )
= ( ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ) ) ).
% sup_Un_eq2
thf(fact_885_sup__Un__eq,axiom,
! [A: $tType,R: set @ A,S3: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R )
@ ^ [X4: A] : ( member @ A @ X4 @ S3 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ R @ S3 ) ) ) ) ).
% sup_Un_eq
thf(fact_886_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A7 )
| ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_887_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_888_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
| ( Q @ X4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_889_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S3: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R )
@ ^ [X4: A] : ( member @ A @ X4 @ S3 ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S3 ) ) ).
% pred_subset_eq
thf(fact_890_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_891_Collect__subset,axiom,
! [A: $tType,A6: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P @ X4 ) ) )
@ A6 ) ).
% Collect_subset
thf(fact_892_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_893_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R ) )
= ( ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ S3 ) ) )
= ( R = S3 ) ) ).
% pred_equals_eq2
thf(fact_894_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $false ) ) ).
% empty_def
thf(fact_895_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_896_insert__compr,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A8: A,B6: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( X4 = A8 )
| ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_897_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_898_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A3 = X4 )
& ( P @ X4 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A3 = X4 )
& ( P @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_899_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A3 )
& ( P @ X4 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A3 )
& ( P @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_900_insert__def,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A8: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] : X4 = A8 ) ) ) ) ).
% insert_def
thf(fact_901_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R )
@ ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ S3 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% pred_subset_eq2
thf(fact_902_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% set_vebt_def
thf(fact_903_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_904_power__numeral__even,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z2: A,W: num] :
( ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ ( bit0 @ W ) ) )
= ( times_times @ A @ ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ W ) ) @ ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_numeral_even
thf(fact_905_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option @ ( product_prod @ nat @ nat ),V2: nat,TreeList: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList @ S ) @ X3 )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( 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 @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_906_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V2: nat,TreeList: list @ vEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList @ Vd ) @ X3 )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( 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 @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_907_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V2: nat,TreeList: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList @ Vc ) @ X3 )
= ( ( X3 = Mi )
| ( X3 = Ma )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_908_vebt__succ_Osimps_I6_J,axiom,
! [X3: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X3 @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X3 )
= ( some @ nat @ Mi ) ) )
& ( ~ ( ord_less @ nat @ X3 @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X3 )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_succ.simps(6)
thf(fact_909_vebt__pred_Osimps_I7_J,axiom,
! [Ma: nat,X3: nat,Mi: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Ma @ X3 )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X3 )
= ( some @ nat @ Ma ) ) )
& ( ~ ( ord_less @ nat @ Ma @ X3 )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X3 )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X3 ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_pred.simps(7)
thf(fact_910_is__succ__in__set__def,axiom,
( vEBT_is_succ_in_set
= ( ^ [Xs: set @ nat,X4: nat,Y3: nat] :
( ( member @ nat @ Y3 @ Xs )
& ( ord_less @ nat @ X4 @ Y3 )
& ! [Z4: nat] :
( ( member @ nat @ Z4 @ Xs )
=> ( ( ord_less @ nat @ X4 @ Z4 )
=> ( ord_less_eq @ nat @ Y3 @ Z4 ) ) ) ) ) ) ).
% is_succ_in_set_def
thf(fact_911_is__pred__in__set__def,axiom,
( vEBT_is_pred_in_set
= ( ^ [Xs: set @ nat,X4: nat,Y3: nat] :
( ( member @ nat @ Y3 @ Xs )
& ( ord_less @ nat @ Y3 @ X4 )
& ! [Z4: nat] :
( ( member @ nat @ Z4 @ Xs )
=> ( ( ord_less @ nat @ Z4 @ X4 )
=> ( ord_less_eq @ nat @ Z4 @ Y3 ) ) ) ) ) ) ).
% is_pred_in_set_def
thf(fact_912_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A8: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A8 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_913_vebt__delete_Osimps_I7_J,axiom,
! [X3: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ( ord_less @ nat @ X3 @ Mi )
| ( ord_less @ nat @ Ma @ X3 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
& ( ~ ( ( ord_less @ nat @ X3 @ Mi )
| ( ord_less @ nat @ Ma @ X3 ) )
=> ( ( ( ( X3 = Mi )
& ( X3 = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
& ( ~ ( ( X3 = Mi )
& ( X3 = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X3 )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X3 = Mi ) @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
@ ( if @ nat
@ ( ( ( X3 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X3 != Mi )
=> ( X3 = Ma ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X3 = Mi ) @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X3 = Mi ) @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
@ ( if @ nat
@ ( ( ( X3 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X3 != Mi )
=> ( X3 = Ma ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).
% vebt_delete.simps(7)
thf(fact_914_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X3 )
= ( ( X3 != Mi )
=> ( ( X3 != Ma )
=> ( ~ ( ord_less @ nat @ X3 @ Mi )
& ( ~ ( ord_less @ nat @ X3 @ Mi )
=> ( ~ ( ord_less @ nat @ Ma @ X3 )
& ( ~ ( ord_less @ nat @ Ma @ X3 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_915_set__swap,axiom,
! [A: $tType,I: nat,Xs2: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs2 @ I @ ( nth @ A @ Xs2 @ J ) ) @ J @ ( nth @ A @ Xs2 @ I ) ) )
= ( set2 @ A @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_916_insert__simp__norm,axiom,
! [X3: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less @ nat @ Mi @ X3 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X3 != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( ord_max @ nat @ X3 @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_norm
thf(fact_917_insert__simp__excp,axiom,
! [Mi: nat,Deg: nat,TreeList: list @ vEBT_VEBT,X3: 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 @ X3 @ Mi )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X3 != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X3 @ ( 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_918_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs2: list @ A,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs2 @ I @ X3 ) @ I )
= X3 ) ) ).
% nth_list_update_eq
thf(fact_919_list__update__beyond,axiom,
! [A: $tType,Xs2: list @ A,I: nat,X3: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ I )
=> ( ( list_update @ A @ Xs2 @ I @ X3 )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_920_length__list__update,axiom,
! [A: $tType,Xs2: list @ A,I: nat,X3: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs2 @ I @ X3 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_list_update
thf(fact_921_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_922_valid__0__not,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_0_not
thf(fact_923_valid__tree__deg__neq__0,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_tree_deg_neq_0
thf(fact_924_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_925_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_926_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_927_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_928_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.right_neutral
thf(fact_929_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_930_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [B2: A,A3: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= A3 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_931_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= A3 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_932_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( plus_plus @ A @ B2 @ A3 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_933_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( plus_plus @ A @ A3 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_934_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X3: A,Y: A] :
( ( ( plus_plus @ A @ X3 @ Y )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_935_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X3: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X3 @ Y ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_936_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add_0
thf(fact_937_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_938_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_zero
thf(fact_939_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_940_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_0_right
thf(fact_941_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_942_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_943_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_944_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_945_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A3 ) ).
% bot_nat_0.extremum
thf(fact_946_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_947_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
= ( ( M2
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_948_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% Nat.add_0_right
thf(fact_949_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus @ nat @ M2 @ M2 )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_950_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_951_mult__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ( times_times @ nat @ M2 @ K2 )
= ( times_times @ nat @ N @ K2 ) )
= ( ( M2 = N )
| ( K2
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_952_mult__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( times_times @ nat @ K2 @ M2 )
= ( times_times @ nat @ K2 @ N ) )
= ( ( M2 = N )
| ( K2
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_953_mult__0__right,axiom,
! [M2: nat] :
( ( times_times @ nat @ M2 @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_954_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_955_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_max @ A @ A3 @ B2 )
= A3 ) ) ) ).
% max.absorb1
thf(fact_956_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_max @ A @ A3 @ B2 )
= B2 ) ) ) ).
% max.absorb2
thf(fact_957_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C3 ) @ A3 )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% max.bounded_iff
thf(fact_958_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X3 )
= X3 ) ) ).
% max_bot
thf(fact_959_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ord_max @ A @ X3 @ ( bot_bot @ A ) )
= X3 ) ) ).
% max_bot2
thf(fact_960_max__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_max @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( suc @ ( ord_max @ nat @ M2 @ N ) ) ) ).
% max_Suc_Suc
thf(fact_961_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_962_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_963_max__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ A3 @ ( zero_zero @ nat ) )
= A3 ) ).
% max_nat.right_neutral
thf(fact_964_max__nat_Oneutr__eq__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( zero_zero @ nat )
= ( ord_max @ nat @ A3 @ B2 ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_965_max__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A3 )
= A3 ) ).
% max_nat.left_neutral
thf(fact_966_max__nat_Oeq__neutr__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_max @ nat @ A3 @ B2 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_967_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_968_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_969_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_970_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_971_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_972_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_973_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_974_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_975_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_976_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_977_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_978_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_979_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_980_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_981_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X3: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_982_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_983_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_984_power__zero__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [K2: num] :
( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ K2 ) )
= ( zero_zero @ A ) ) ) ).
% power_zero_numeral
thf(fact_985_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_986_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_987_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_988_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_989_max__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X3 ) )
= ( numeral_numeral @ A @ X3 ) ) ) ).
% max_0_1(3)
thf(fact_990_max__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X3 ) @ ( zero_zero @ A ) )
= ( numeral_numeral @ A @ X3 ) ) ) ).
% max_0_1(4)
thf(fact_991_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% add_gr_0
thf(fact_992_one__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( times_times @ nat @ M2 @ N ) )
= ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% one_eq_mult_iff
thf(fact_993_mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times @ nat @ M2 @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% mult_eq_1_iff
thf(fact_994_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_995_max__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X3 ) )
= ( numeral_numeral @ A @ X3 ) ) ) ).
% max_0_1(5)
thf(fact_996_max__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X3 ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ X3 ) ) ) ).
% max_0_1(6)
thf(fact_997_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_998_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_999_mult__less__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M2 @ K2 ) @ ( times_times @ nat @ N @ K2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
& ( ord_less @ nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1000_nat__power__eq__Suc__0__iff,axiom,
! [X3: nat,M2: nat] :
( ( ( power_power @ nat @ X3 @ M2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ( X3
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1001_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_1002_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( minus_minus @ nat @ M2 @ N )
= ( zero_zero @ nat ) ) ) ).
% diff_is_0_eq'
thf(fact_1003_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1004_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_1005_nat__zero__less__power__iff,axiom,
! [X3: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X3 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1006_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) )
= A3 )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1007_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,W: num] :
( ( A3
= ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) )
= B2 ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1008_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C3 ) @ A3 ) @ B2 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self4
thf(fact_1009_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C3 @ B2 ) @ A3 ) @ B2 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self3
thf(fact_1010_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) @ B2 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self2
thf(fact_1011_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C3 @ B2 ) ) @ B2 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self1
thf(fact_1012_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_1013_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_1014_one__le__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M2 @ N ) )
= ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
& ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1015_mult__le__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M2 @ K2 ) @ ( times_times @ nat @ N @ K2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1016_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1017_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ M2 ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less @ nat @ N @ M2 ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1018_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_1019_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_1020_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_1021_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_1022_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_1023_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_1024_power2__eq__iff__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( X3 = Y ) ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1025_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ M2 ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less_eq @ nat @ N @ M2 ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_1026_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_1027_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_1028_option_Osize__neq,axiom,
! [A: $tType,X3: option @ A] :
( ( size_size @ ( option @ A ) @ X3 )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_1029_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_1030_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X3: A] :
( ( ( zero_zero @ A )
= X3 )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_1031_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_1032_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A3: A,D3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ( ord_less_eq @ A @ D3 @ B2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C3 @ D3 ) @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ) ).
% max.mono
thf(fact_1033_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3
= ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.orderE
thf(fact_1034_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( ord_max @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% max.orderI
thf(fact_1035_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C3 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A3 )
=> ~ ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% max.boundedE
thf(fact_1036_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C3 ) @ A3 ) ) ) ) ).
% max.boundedI
thf(fact_1037_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B8: A,A8: A] :
( A8
= ( ord_max @ A @ A8 @ B8 ) ) ) ) ) ).
% max.order_iff
thf(fact_1038_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.cobounded1
thf(fact_1039_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.cobounded2
thf(fact_1040_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less_eq @ A @ Z2 @ ( ord_max @ A @ X3 @ Y ) )
= ( ( ord_less_eq @ A @ Z2 @ X3 )
| ( ord_less_eq @ A @ Z2 @ Y ) ) ) ) ).
% le_max_iff_disj
thf(fact_1041_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B8: A,A8: A] :
( ( ord_max @ A @ A8 @ B8 )
= A8 ) ) ) ) ).
% max.absorb_iff1
thf(fact_1042_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] :
( ( ord_max @ A @ A8 @ B8 )
= B8 ) ) ) ) ).
% max.absorb_iff2
thf(fact_1043_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ C3 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.coboundedI1
thf(fact_1044_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.coboundedI2
thf(fact_1045_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A8: A,B8: A] : ( if @ A @ ( ord_less_eq @ A @ A8 @ B8 ) @ B8 @ A8 ) ) ) ) ).
% max_def
thf(fact_1046_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( ord_max @ A @ X3 @ Y )
= X3 ) ) ) ).
% max_absorb1
thf(fact_1047_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_max @ A @ X3 @ Y )
= Y ) ) ) ).
% max_absorb2
thf(fact_1048_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X3 @ Y ) @ Z2 )
= ( ord_max @ A @ ( plus_plus @ A @ X3 @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ).
% max_add_distrib_left
thf(fact_1049_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( plus_plus @ A @ X3 @ ( ord_max @ A @ Y @ Z2 ) )
= ( ord_max @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( plus_plus @ A @ X3 @ Z2 ) ) ) ) ).
% max_add_distrib_right
thf(fact_1050_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X3 @ Y ) @ Z2 )
= ( ord_max @ A @ ( minus_minus @ A @ X3 @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ).
% max_diff_distrib_left
thf(fact_1051_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_1052_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_1053_nat__add__max__left,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M2 @ N ) @ Q3 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M2 @ Q3 ) @ ( plus_plus @ nat @ N @ Q3 ) ) ) ).
% nat_add_max_left
thf(fact_1054_nat__add__max__right,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( plus_plus @ nat @ M2 @ ( ord_max @ nat @ N @ Q3 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( plus_plus @ nat @ M2 @ Q3 ) ) ) ).
% nat_add_max_right
thf(fact_1055_nat__mult__max__right,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( times_times @ nat @ M2 @ ( ord_max @ nat @ N @ Q3 ) )
= ( ord_max @ nat @ ( times_times @ nat @ M2 @ N ) @ ( times_times @ nat @ M2 @ Q3 ) ) ) ).
% nat_mult_max_right
thf(fact_1056_nat__mult__max__left,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M2 @ N ) @ Q3 )
= ( ord_max @ nat @ ( times_times @ nat @ M2 @ Q3 ) @ ( times_times @ nat @ N @ Q3 ) ) ) ).
% nat_mult_max_left
thf(fact_1057_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ).
% zero_le
thf(fact_1058_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_1059_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_1060_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [M2: A,N: A] :
( ( ord_less @ A @ M2 @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_1061_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_1062_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_1063_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_1064_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_1065_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_1066_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_1067_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [A8: A,B8: A] :
( ( minus_minus @ A @ A8 @ B8 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1068_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_1069_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_1070_vebt__buildup_Ocases,axiom,
! [X3: nat] :
( ( X3
!= ( zero_zero @ nat ) )
=> ( ( X3
!= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ! [Va3: nat] :
( X3
!= ( suc @ ( suc @ Va3 ) ) ) ) ) ).
% vebt_buildup.cases
thf(fact_1071_list__decode_Ocases,axiom,
! [X3: nat] :
( ( X3
!= ( zero_zero @ nat ) )
=> ~ ! [N2: nat] :
( X3
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_1072_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( ( zero_zero @ nat )
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_1073_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_1074_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1075_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_1076_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat4: nat] :
( Y
!= ( suc @ Nat4 ) ) ) ).
% old.nat.exhaust
thf(fact_1077_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1078_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X5: nat] : ( P @ X5 @ ( zero_zero @ nat ) )
=> ( ! [Y4: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y4 ) )
=> ( ! [X5: nat,Y4: nat] :
( ( P @ X5 @ Y4 )
=> ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_1079_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_1080_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_1081_Zero__neq__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_1082_Zero__not__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_1083_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% not0_implies_Suc
thf(fact_1084_infinite__descent0__measure,axiom,
! [A: $tType,V3: A > nat,P: A > $o,X3: A] :
( ! [X5: A] :
( ( ( V3 @ X5 )
= ( zero_zero @ nat ) )
=> ( P @ X5 ) )
=> ( ! [X5: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V3 @ X5 ) )
=> ( ~ ( P @ X5 )
=> ? [Y6: A] :
( ( ord_less @ nat @ ( V3 @ Y6 ) @ ( V3 @ X5 ) )
& ~ ( P @ Y6 ) ) ) )
=> ( P @ X3 ) ) ) ).
% infinite_descent0_measure
thf(fact_1085_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1086_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_1087_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_1088_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_1089_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_1090_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_1091_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_1092_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1093_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_1094_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_1095_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_1096_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1097_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus @ nat @ M2 @ N )
= M2 )
=> ( N
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_1098_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N @ M2 )
= ( zero_zero @ nat ) )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1099_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1100_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_1101_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ( power_power @ A @ A3 @ N )
= ( power_power @ A @ B2 @ N ) )
= ( A3 = B2 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1102_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( ( power_power @ A @ A3 @ N )
= ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A3 = B2 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1103_nat__minus__add__max,axiom,
! [N: nat,M2: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ N @ M2 ) @ M2 )
= ( ord_max @ nat @ N @ M2 ) ) ).
% nat_minus_add_max
thf(fact_1104_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_1105_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_1106_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_1107_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1108_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1109_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1110_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1111_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1112_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1113_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_1114_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_1115_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_right_mono
thf(fact_1116_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1117_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_left_mono
thf(fact_1118_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1119_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1120_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% split_mult_pos_le
thf(fact_1121_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_1122_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ( 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 @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1123_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_1124_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_1125_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_1126_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_1127_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% add_increasing
thf(fact_1128_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_1129_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% add_increasing2
thf(fact_1130_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1131_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_1132_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X3 @ Y )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1133_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X3 @ Y )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1134_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_1135_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_1136_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_1137_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_add_strict
thf(fact_1138_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ! [C2: A] :
( ( B2
= ( plus_plus @ A @ A3 @ C2 ) )
=> ( C2
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1139_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_pos_pos
thf(fact_1140_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_1141_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X3 @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_1142_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A8 @ B8 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_1143_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A8: A,B8: A] : ( ord_less @ A @ ( minus_minus @ A @ A8 @ B8 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_1144_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono
thf(fact_1145_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_1146_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_1147_length__pos__if__in__set,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1148_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_1149_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less @ nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1150_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% gr0_implies_Suc
thf(fact_1151_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_1152_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1153_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_1154_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus @ nat @ M2 @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M2
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% add_is_1
thf(fact_1155_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( plus_plus @ nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M2
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% one_is_add
thf(fact_1156_option_Osize_I4_J,axiom,
! [A: $tType,X2: A] :
( ( size_size @ ( option @ A ) @ ( some @ A @ X2 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(4)
thf(fact_1157_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K: nat] :
( ( ord_less_eq @ nat @ K @ N )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ~ ( P @ I2 ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_1158_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_1159_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ? [K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ( plus_plus @ nat @ I @ K )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1160_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_1161_mult__less__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less @ nat @ ( times_times @ nat @ K2 @ I ) @ ( times_times @ nat @ K2 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1162_mult__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less @ nat @ ( times_times @ nat @ I @ K2 ) @ ( times_times @ nat @ J @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_1163_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_1164_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M2 ) )
= ( zero_zero @ nat ) ) ).
% diff_add_0
thf(fact_1165_nat__power__less__imp__less,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ ( power_power @ nat @ I @ M2 ) @ ( power_power @ nat @ I @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1166_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times @ nat @ M2 @ N ) )
=> ( ( N
= ( one_one @ nat ) )
| ( M2
= ( zero_zero @ nat ) ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1167_vebt__member_Osimps_I3_J,axiom,
! [V2: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X3 ) ).
% vebt_member.simps(3)
thf(fact_1168_vebt__insert_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) @ X3 )
= ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) ) ).
% vebt_insert.simps(2)
thf(fact_1169_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Ux ) ).
% VEBT_internal.naive_member.simps(2)
thf(fact_1170_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1171_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1172_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_1173_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1174_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1175_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_1176_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ( 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 @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1177_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1178_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1179_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1180_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_1181_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_1182_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ( 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 @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1183_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ( 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 @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1184_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_1185_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_1186_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_1187_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_1188_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_1189_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_1190_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_1191_div__positive,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_positive
thf(fact_1192_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1193_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_1194_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y @ X3 ) @ X3 ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1195_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X3 @ Y ) @ X3 ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1196_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_1197_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_1198_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_1199_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) )
= ( ( X3
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1200_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X3: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_1201_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_1202_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_1203_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_1204_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,W: num] :
( ( ( divide_divide @ A @ B2 @ C3 )
= ( numeral_numeral @ A @ W ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_1205_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B2: A,C3: A] :
( ( ( numeral_numeral @ A @ W )
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 )
= B2 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_1206_vebt__member_Osimps_I4_J,axiom,
! [V2: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X3 ) ).
% vebt_member.simps(4)
thf(fact_1207_power__inject__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( power_power @ A @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( A3 = B2 ) ) ) ) ) ).
% power_inject_base
thf(fact_1208_power__le__imp__le__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( power_power @ A @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% power_le_imp_le_base
thf(fact_1209_div__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_1210_div__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_1211_vebt__delete_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,TrLst: list @ vEBT_VEBT,Smry: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) ) ).
% vebt_delete.simps(5)
thf(fact_1212_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1213_num_Osize_I5_J,axiom,
! [X2: num] :
( ( size_size @ num @ ( bit0 @ X2 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(5)
thf(fact_1214_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs3: list @ A] :
( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_1215_neq__if__length__neq,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_1216_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K: nat] :
( ( ord_less @ nat @ K @ N )
& ! [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ K )
=> ~ ( P @ I2 ) )
& ( P @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1217_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_1218_one__less__mult,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M2 @ N ) ) ) ) ).
% one_less_mult
thf(fact_1219_n__less__m__mult__n,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ord_less @ nat @ N @ ( times_times @ nat @ M2 @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1220_n__less__n__mult__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ord_less @ nat @ N @ ( times_times @ nat @ N @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1221_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1222_power__gt__expt,axiom,
! [N: nat,K2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ K2 @ ( power_power @ nat @ N @ K2 ) ) ) ).
% power_gt_expt
thf(fact_1223_nat__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1224_div__le__mono2,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ K2 @ N ) @ ( divide_divide @ nat @ K2 @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1225_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M2 @ N ) )
= ( ( ord_less_eq @ nat @ N @ M2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1226_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_1227_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( ( ( ord_less @ nat @ A3 @ B2 )
=> ( P @ ( zero_zero @ nat ) ) )
& ! [D5: nat] :
( ( A3
= ( plus_plus @ nat @ B2 @ D5 ) )
=> ( P @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1228_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( ~ ( ( ( ord_less @ nat @ A3 @ B2 )
& ~ ( P @ ( zero_zero @ nat ) ) )
| ? [D5: nat] :
( ( A3
= ( plus_plus @ nat @ B2 @ D5 ) )
& ~ ( P @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1229_vebt__insert_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) @ X3 )
= ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) ) ).
% vebt_insert.simps(3)
thf(fact_1230_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Uz ) ).
% VEBT_internal.membermima.simps(2)
thf(fact_1231_vebt__delete_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,Tr: list @ vEBT_VEBT,Sm: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) ) ).
% vebt_delete.simps(6)
thf(fact_1232_mult__le__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1233_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_1234_mult__le__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1235_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_1236_mult__less__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1237_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_1238_mult__less__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1239_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_1240_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X3: A,A3: A,Y: A,U: A,V2: A] :
( ( ord_less_eq @ A @ X3 @ 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 @ X3 ) @ ( times_times @ A @ V2 @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1241_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C3: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_1242_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_1243_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_1244_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_1245_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_1246_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N5: nat,A3: A] :
( ( ord_less @ nat @ N @ N5 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N5 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1247_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N5: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N5 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_1248_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_1249_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_1250_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_1251_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_1252_pos2,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).
% pos2
thf(fact_1253_power__diff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,N: nat,M2: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( power_power @ A @ A3 @ ( minus_minus @ nat @ M2 @ N ) )
= ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_diff
thf(fact_1254_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M5: nat,N3: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M5 @ N3 )
| ( N3
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M5 @ N3 ) @ N3 ) ) ) ) ) ).
% div_if
thf(fact_1255_div__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ~ ( ord_less @ nat @ M2 @ N )
=> ( ( divide_divide @ nat @ M2 @ N )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ) ) ).
% div_geq
thf(fact_1256_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_1257_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( minus_minus @ nat @ ( suc @ M2 ) @ N )
= ( minus_minus @ nat @ M2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1258_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q3 )
=> ( ( ord_less_eq @ nat @ M2 @ ( divide_divide @ nat @ N @ Q3 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M2 @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1259_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M5: nat,N3: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N3
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1260_split__div,axiom,
! [P: nat > $o,M2: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I4 ) @ J3 ) )
=> ( P @ I4 ) ) ) ) ) ) ).
% split_div
thf(fact_1261_dividend__less__div__times,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( divide_divide @ nat @ M2 @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1262_dividend__less__times__div,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ N @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M2 @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1263_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M5: nat,N3: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N3 @ ( times_times @ nat @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1264_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va: list @ vEBT_VEBT,Vb: vEBT_VEBT,X3: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ Va @ Vb ) @ X3 )
= ( ( X3 = Mi )
| ( X3 = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_1265_vebt__pred_Osimps_I5_J,axiom,
! [V2: product_prod @ nat @ nat,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Vd @ Ve ) @ Vf )
= ( none @ nat ) ) ).
% vebt_pred.simps(5)
thf(fact_1266_vebt__succ_Osimps_I4_J,axiom,
! [V2: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Vc @ Vd ) @ Ve )
= ( none @ nat ) ) ).
% vebt_succ.simps(4)
thf(fact_1267_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_1268_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ 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 @ W ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_1269_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X3: A,A3: A,Y: A,U: A,V2: A] :
( ( ord_less @ A @ X3 @ 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 @ X3 ) @ ( times_times @ A @ V2 @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1270_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_1271_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_1272_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_1273_power2__eq__imp__eq,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,Y: A] :
( ( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( X3 = Y ) ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_1274_power2__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ).
% power2_le_imp_le
thf(fact_1275_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_1276_exp__add__not__zero__imp__left,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_left
thf(fact_1277_exp__add__not__zero__imp__right,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_right
thf(fact_1278_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,M2: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) )
!= ( zero_zero @ A ) ) ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_1279_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,N: nat,M2: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
= ( power_power @ A @ A3 @ ( minus_minus @ nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1280_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_1281_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_1282_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1283_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P5: A,M5: nat] :
( if @ A
@ ( M5
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P5 @ ( power_power @ A @ P5 @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1284_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_1285_le__div__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ nat @ M2 @ N )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_1286_split__div_H,axiom,
! [P: nat > $o,M2: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
& ( P @ ( zero_zero @ nat ) ) )
| ? [Q4: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q4 ) @ M2 )
& ( ord_less @ nat @ M2 @ ( times_times @ nat @ N @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_1287_vebt__pred_Osimps_I6_J,axiom,
! [V2: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Vj )
= ( none @ nat ) ) ).
% vebt_pred.simps(6)
thf(fact_1288_vebt__succ_Osimps_I5_J,axiom,
! [V2: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Vi )
= ( none @ nat ) ) ).
% vebt_succ.simps(5)
thf(fact_1289_power2__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ X3 @ Y ) ) ) ) ).
% power2_less_imp_less
thf(fact_1290_sum__power2__ge__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_1291_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_1292_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_1293_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X3
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_1294_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_1295_nat__bit__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_1296_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_1297_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_1298_subset__code_I1_J,axiom,
! [A: $tType,Xs2: list @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ B5 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( member @ A @ X4 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1299_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs2: list @ A] :
( ! [Xs3: list @ A] :
( ! [Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_1300_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_1301_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_1302_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X3: nat,N: nat,M2: nat] :
( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M2 ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_1303_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X3: nat,N: nat,M2: nat] :
( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M2 ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( vEBT_VEBT_low @ X3 @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_1304_arith__geo__mean,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,X3: A,Y: A] :
( ( ( power_power @ A @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ X3 @ Y ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ U @ ( divide_divide @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_1305_nth__equalityI,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I3 )
= ( nth @ A @ Ys @ I3 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_1306_Skolem__list__nth,axiom,
! [A: $tType,K2: nat,P: nat > A > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K2 )
=> ? [X8: A] : ( P @ I4 @ X8 ) ) )
= ( ? [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= K2 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K2 )
=> ( P @ I4 @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1307_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z: list @ A] : Y5 = Z )
= ( ^ [Xs: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I4 )
= ( nth @ A @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1308_set__update__subsetI,axiom,
! [A: $tType,Xs2: list @ A,A6: set @ A,X3: A,I: nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs2 @ I @ X3 ) ) @ A6 ) ) ) ).
% set_update_subsetI
thf(fact_1309_VEBT__internal_OminNull_Osimps_I5_J,axiom,
! [Uz: product_prod @ nat @ nat,Va: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va @ Vb @ Vc ) ) ).
% VEBT_internal.minNull.simps(5)
thf(fact_1310_vebt__delete_Osimps_I4_J,axiom,
! [Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Uu2 )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) ) ).
% vebt_delete.simps(4)
thf(fact_1311_vebt__member_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X3 ) ).
% vebt_member.simps(2)
thf(fact_1312_VEBT__internal_OminNull_Osimps_I4_J,axiom,
! [Uw2: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux @ Uy ) ) ).
% VEBT_internal.minNull.simps(4)
thf(fact_1313_all__set__conv__all__nth,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X4 ) ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1314_all__nth__imp__all__set,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o,X3: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ I3 ) ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1315_in__set__conv__nth,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ( nth @ A @ Xs2 @ I4 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1316_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs2: list @ A,P: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X5 ) )
=> ( P @ ( nth @ A @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1317_nth__mem,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( member @ A @ ( nth @ A @ Xs2 @ N ) @ ( set2 @ A @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1318_set__update__subset__insert,axiom,
! [A: $tType,Xs2: list @ A,I: nat,X3: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs2 @ I @ X3 ) ) @ ( insert2 @ A @ X3 @ ( set2 @ A @ Xs2 ) ) ) ).
% set_update_subset_insert
thf(fact_1319_set__update__memI,axiom,
! [A: $tType,N: nat,Xs2: list @ A,X3: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( member @ A @ X3 @ ( set2 @ A @ ( list_update @ A @ Xs2 @ N @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_1320_nth__list__update,axiom,
! [A: $tType,I: nat,Xs2: list @ A,J: nat,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( I = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs2 @ I @ X3 ) @ J )
= X3 ) )
& ( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs2 @ I @ X3 ) @ J )
= ( nth @ A @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_1321_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs2: list @ A,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( list_update @ A @ Xs2 @ I @ X3 )
= Xs2 )
= ( ( nth @ A @ Xs2 @ I )
= X3 ) ) ) ).
% list_update_same_conv
thf(fact_1322_vebt__insert_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X3 )
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X3 = Mi )
| ( X3 = Ma ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ X3 @ Mi ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ).
% vebt_insert.simps(5)
thf(fact_1323_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1324_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1325_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_1326_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_1327_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_1328_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_1329_vebt__pred_Oelims,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: option @ nat] :
( ( ( vEBT_vebt_pred @ X3 @ Xa2 )
= Y )
=> ( ( ? [Uu: $o,Uv: $o] :
( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
!= ( none @ nat ) ) ) )
=> ( ! [A5: $o] :
( ? [Uw: $o] :
( X3
= ( vEBT_Leaf @ A5 @ Uw ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( ( A5
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ? [Va3: nat] :
( Xa2
= ( suc @ ( suc @ Va3 ) ) )
=> ~ ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A5
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y
= ( none @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
=> ~ ( ( ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( Y
= ( some @ nat @ Ma2 ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi2 @ Xa2 ) @ ( some @ nat @ Mi2 ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.elims
thf(fact_1330_vebt__succ_Oelims,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: option @ nat] :
( ( ( vEBT_vebt_succ @ X3 @ Xa2 )
= Y )
=> ( ! [Uu: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ~ ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ( ( ? [Uv: $o,Uw: $o] :
( X3
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ( ? [N2: nat] :
( Xa2
= ( suc @ N2 ) )
=> ( Y
!= ( none @ nat ) ) ) )
=> ( ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
=> ~ ( ( ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( Y
= ( some @ nat @ Mi2 ) ) )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.elims
thf(fact_1331_vebt__delete_Oelims,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X3 @ Xa2 )
= Y )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ B4 ) ) ) )
=> ( ! [A5: $o] :
( ? [B4: $o] :
( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( vEBT_Leaf @ A5 @ $false ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ? [N2: nat] :
( Xa2
= ( suc @ ( suc @ N2 ) ) )
=> ( Y
!= ( vEBT_Leaf @ A5 @ B4 ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( Y
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary3 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
=> ~ ( ( ( ( ord_less @ nat @ Xa2 @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa2 ) )
=> ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) ) )
& ( ~ ( ( ord_less @ nat @ Xa2 @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa2 ) )
=> ( ( ( ( Xa2 = Mi2 )
& ( Xa2 = Ma2 ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) ) )
& ( ~ ( ( Xa2 = Mi2 )
& ( Xa2 = Ma2 ) )
=> ( Y
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa2 = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa2 != Mi2 )
=> ( Xa2 = Ma2 ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( Xa2 = Mi2 ) @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va3 ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa2 = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa2 != Mi2 )
=> ( Xa2 = Ma2 ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va3 ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary3 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_delete.elims
thf(fact_1332_Leaf__0__not,axiom,
! [A3: $o,B2: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A3 @ B2 ) @ ( zero_zero @ nat ) ) ).
% Leaf_0_not
thf(fact_1333_deg__1__Leafy,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( N
= ( one_one @ nat ) )
=> ? [A5: $o,B4: $o] :
( T2
= ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ).
% deg_1_Leafy
thf(fact_1334_deg__1__Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
=> ? [A5: $o,B4: $o] :
( T2
= ( vEBT_Leaf @ A5 @ B4 ) ) ) ).
% deg_1_Leaf
thf(fact_1335_deg1Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
= ( ? [A8: $o,B8: $o] :
( T2
= ( vEBT_Leaf @ A8 @ B8 ) ) ) ) ).
% deg1Leaf
thf(fact_1336_VEBT_Oinject_I2_J,axiom,
! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
( ( ( vEBT_Leaf @ X21 @ X222 )
= ( vEBT_Leaf @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% VEBT.inject(2)
thf(fact_1337_half__nonnegative__int__iff,axiom,
! [K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 ) ) ).
% half_nonnegative_int_iff
thf(fact_1338_half__negative__int__iff,axiom,
! [K2: int] :
( ( ord_less @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% half_negative_int_iff
thf(fact_1339_VEBT_Osize_I4_J,axiom,
! [X21: $o,X222: $o] :
( ( size_size @ vEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size(4)
thf(fact_1340_VEBT_Odistinct_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leaf @ X21 @ X222 ) ) ).
% VEBT.distinct(1)
thf(fact_1341_VEBT_Oexhaust,axiom,
! [Y: vEBT_VEBT] :
( ! [X112: option @ ( product_prod @ nat @ nat ),X122: nat,X132: list @ vEBT_VEBT,X142: vEBT_VEBT] :
( Y
!= ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X223: $o] :
( Y
!= ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% VEBT.exhaust
thf(fact_1342_VEBT__internal_Oinsert_H_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [A5: $o,B4: $o,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ X5 ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ X5 ) ) ) ).
% VEBT_internal.insert'.cases
thf(fact_1343_VEBT__internal_OminNull_Osimps_I1_J,axiom,
vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% VEBT_internal.minNull.simps(1)
thf(fact_1344_VEBT__internal_OminNull_Osimps_I2_J,axiom,
! [Uv2: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv2 ) ) ).
% VEBT_internal.minNull.simps(2)
thf(fact_1345_VEBT__internal_OminNull_Osimps_I3_J,axiom,
! [Uu2: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu2 @ $true ) ) ).
% VEBT_internal.minNull.simps(3)
thf(fact_1346_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o,Uw2: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) ).
% VEBT_internal.membermima.simps(1)
thf(fact_1347_vebt__delete_Osimps_I3_J,axiom,
! [A3: $o,B2: $o,N: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ N ) ) )
= ( vEBT_Leaf @ A3 @ B2 ) ) ).
% vebt_delete.simps(3)
thf(fact_1348_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_1349_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o,D3: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D3 )
= ( D3
= ( one_one @ nat ) ) ) ).
% VEBT_internal.valid'.simps(1)
thf(fact_1350_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
! [A3: $o,B2: $o,X3: nat] :
( ( vEBT_VEBT_insert @ ( vEBT_Leaf @ A3 @ B2 ) @ X3 )
= ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B2 ) @ X3 ) ) ).
% VEBT_internal.insert'.simps(1)
thf(fact_1351_VEBT__internal_Onaive__member_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [A5: $o,B4: $o,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ X5 ) )
=> ( ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,Ux2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Ux2 ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) @ X5 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_1352_invar__vebt_Ointros_I1_J,axiom,
! [A3: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ).
% invar_vebt.intros(1)
thf(fact_1353_vebt__delete_Osimps_I2_J,axiom,
! [A3: $o,B2: $o] :
( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ A3 @ $false ) ) ).
% vebt_delete.simps(2)
thf(fact_1354_VEBT__internal_OminNull_Ocases,axiom,
! [X3: vEBT_VEBT] :
( ( X3
!= ( vEBT_Leaf @ $false @ $false ) )
=> ( ! [Uv: $o] :
( X3
!= ( vEBT_Leaf @ $true @ Uv ) )
=> ( ! [Uu: $o] :
( X3
!= ( vEBT_Leaf @ Uu @ $true ) )
=> ( ! [Uw: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.cases
thf(fact_1355_vebt__member_Osimps_I1_J,axiom,
! [A3: $o,B2: $o,X3: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A3 @ B2 ) @ X3 )
= ( ( ( X3
= ( zero_zero @ nat ) )
=> A3 )
& ( ( X3
!= ( zero_zero @ nat ) )
=> ( ( ( X3
= ( one_one @ nat ) )
=> B2 )
& ( X3
= ( one_one @ nat ) ) ) ) ) ) ).
% vebt_member.simps(1)
thf(fact_1356_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_1357_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X3 )
=> ( ! [Uv: $o] :
( X3
!= ( vEBT_Leaf @ $true @ Uv ) )
=> ( ! [Uu: $o] :
( X3
!= ( vEBT_Leaf @ Uu @ $true ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% VEBT_internal.minNull.elims(3)
thf(fact_1358_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A3: $o,B2: $o,X3: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A3 @ B2 ) @ X3 )
= ( ( ( X3
= ( zero_zero @ nat ) )
=> A3 )
& ( ( X3
!= ( zero_zero @ nat ) )
=> ( ( ( X3
= ( one_one @ nat ) )
=> B2 )
& ( X3
= ( one_one @ nat ) ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_1359_vebt__succ_Osimps_I2_J,axiom,
! [Uv2: $o,Uw2: $o,N: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) )
= ( none @ nat ) ) ).
% vebt_succ.simps(2)
thf(fact_1360_vebt__pred_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o] :
( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) )
= ( none @ nat ) ) ).
% vebt_pred.simps(1)
thf(fact_1361_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X3 )
=> ( ( X3
!= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ! [Uw: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_1362_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_1363_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_1364_real__arch__pow__inv,axiom,
! [Y: real,X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ? [N2: nat] : ( ord_less @ real @ ( power_power @ real @ X3 @ N2 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1365_vebt__mint_Ocases,axiom,
! [X3: vEBT_VEBT] :
( ! [A5: $o,B4: $o] :
( X3
!= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% vebt_mint.cases
thf(fact_1366_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X3 )
= Y )
=> ( ( ( X3
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y )
=> ( ( ? [Uv: $o] :
( X3
= ( vEBT_Leaf @ $true @ Uv ) )
=> Y )
=> ( ( ? [Uu: $o] :
( X3
= ( vEBT_Leaf @ Uu @ $true ) )
=> Y )
=> ( ( ? [Uw: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) )
=> ~ Y )
=> ~ ( ? [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> Y ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_1367_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_1368_realpow__pos__nth__unique,axiom,
! [N: nat,A3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [X5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X5 )
& ( ( power_power @ real @ X5 @ N )
= A3 )
& ! [Y6: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y6 )
& ( ( power_power @ real @ Y6 @ N )
= A3 ) )
=> ( Y6 = X5 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1369_neg__zdiv__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1370_pos__zdiv__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ B2 @ A3 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1371_vebt__mint_Osimps_I1_J,axiom,
! [A3: $o,B2: $o] :
( ( A3
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A3 @ B2 ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A3
=> ( ( B2
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A3 @ B2 ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A3 @ B2 ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_mint.simps(1)
thf(fact_1372_vebt__maxt_Osimps_I1_J,axiom,
! [B2: $o,A3: $o] :
( ( B2
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A3 @ B2 ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( A3
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A3 @ B2 ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A3
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A3 @ B2 ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_maxt.simps(1)
thf(fact_1373_vebt__pred_Osimps_I2_J,axiom,
! [A3: $o,Uw2: $o] :
( ( A3
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A3
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( none @ nat ) ) ) ) ).
% vebt_pred.simps(2)
thf(fact_1374_vebt__succ_Osimps_I1_J,axiom,
! [B2: $o,Uu2: $o] :
( ( B2
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu2 @ B2 ) @ ( zero_zero @ nat ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu2 @ B2 ) @ ( zero_zero @ nat ) )
= ( none @ nat ) ) ) ) ).
% vebt_succ.simps(1)
thf(fact_1375_int__power__div__base,axiom,
! [M2: nat,K2: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( divide_divide @ int @ ( power_power @ int @ K2 @ M2 ) @ K2 )
= ( power_power @ int @ K2 @ ( minus_minus @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_1376_vebt__pred_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu: $o,Uv: $o] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) ) )
=> ( ! [A5: $o,Uw: $o] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A5: $o,B4: $o,Va3: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( suc @ Va3 ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) @ Vb2 ) )
=> ( ! [V: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Vf2 ) )
=> ( ! [V: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Vj2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ X5 ) ) ) ) ) ) ) ) ).
% vebt_pred.cases
thf(fact_1377_vebt__succ_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu: $o,B4: $o] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ B4 ) @ ( zero_zero @ nat ) ) )
=> ( ! [Uv: $o,Uw: $o,N2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) )
=> ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Va2 ) )
=> ( ! [V: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) @ Ve2 ) )
=> ( ! [V: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Vi2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ X5 ) ) ) ) ) ) ) ).
% vebt_succ.cases
thf(fact_1378_vebt__insert_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [A5: $o,B4: $o,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ X5 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) @ X5 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) @ X5 ) )
=> ( ! [V: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary3 ) @ X5 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ X5 ) ) ) ) ) ) ).
% vebt_insert.cases
thf(fact_1379_VEBT__internal_Omembermima_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu: $o,Uv: $o,Uw: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Uz2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ X5 ) )
=> ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) @ X5 ) )
=> ~ ! [V: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd2 ) @ X5 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_1380_vebt__delete_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [A5: $o,B4: $o] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( zero_zero @ nat ) ) )
=> ( ! [A5: $o,B4: $o] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A5: $o,B4: $o,N2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( suc @ N2 ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT,Uu: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary3 ) @ Uu ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) @ X5 ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) @ X5 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ X5 ) ) ) ) ) ) ) ) ).
% vebt_delete.cases
thf(fact_1381_vebt__member_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [A5: $o,B4: $o,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ X5 ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X5 ) )
=> ( ! [V: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ X5 ) )
=> ( ! [V: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ X5 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ X5 ) ) ) ) ) ) ).
% vebt_member.cases
thf(fact_1382_vebt__pred_Osimps_I3_J,axiom,
! [B2: $o,A3: $o,Va: nat] :
( ( B2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( A3
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A3
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_pred.simps(3)
thf(fact_1383_add__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ).
% add_divide_distrib
thf(fact_1384_VEBT__internal_Oinsert_H_Oelims,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_VEBT_insert @ X3 @ Xa2 )
= Y )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y
!= ( vEBT_vebt_insert @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) )
=> ~ ( ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) @ Xa2 )
=> ( Y
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) ) )
& ( ~ ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) @ Xa2 )
=> ( Y
= ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.insert'.elims
thf(fact_1385_vebt__mint_Oelims,axiom,
! [X3: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_mint @ X3 )
= Y )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ( ( A5
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat] :
( ? [Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( some @ nat @ Mi2 ) ) ) ) ) ) ).
% vebt_mint.elims
thf(fact_1386_vebt__maxt_Oelims,axiom,
! [X3: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_maxt @ X3 )
= Y )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A5
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat] :
( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( some @ nat @ Ma2 ) ) ) ) ) ) ).
% vebt_maxt.elims
thf(fact_1387_divide__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_le_0_iff
thf(fact_1388_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% divide_right_mono
thf(fact_1389_zero__le__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_divide_iff
thf(fact_1390_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1391_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1392_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1393_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1394_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( divide_divide @ A @ A3 @ C3 ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_1395_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ Y @ E2 ) ) )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% field_le_epsilon
thf(fact_1396_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_1397_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1398_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1399_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_1400_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_1401_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A,W: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less @ A @ W @ Z2 )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_1402_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A,W: A,Z2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z2 )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% frac_less
thf(fact_1403_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X3: A,W: A,Z2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% frac_le
thf(fact_1404_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= B2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1405_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= A3 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1406_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X3: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_1407_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,X3: A,Z2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Y ) @ Z2 )
= ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% add_frac_num
thf(fact_1408_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X3: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ X3 @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% add_num_frac
thf(fact_1409_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X3 @ ( divide_divide @ A @ Y @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ) ).
% add_divide_eq_iff
thf(fact_1410_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% divide_add_eq_iff
thf(fact_1411_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ A3 @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_1412_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).
% gt_half_sum
thf(fact_1413_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_1414_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_1415_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
= Y )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y
= ( ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) )
=> Y )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_1416_vebt__member_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X3 @ Xa2 )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary3: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_1417_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
= Y )
=> ( ( ? [Uu: $o,Uv: $o] :
( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> Y )
=> ( ( ? [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> Y )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ( Y
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
=> ( Y
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) )
=> ~ ! [V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd2 ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_1418_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
=> ( ! [Uu: $o,Uv: $o] :
( X3
!= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_1419_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ! [Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less @ A @ Z3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X3 ) @ Y ) ) )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% field_le_mult_one_interval
thf(fact_1420_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_1421_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ Y ) @ X3 )
=> ( ord_less_eq @ A @ Z2 @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1422_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X3: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ Z2 ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1423_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_1424_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_1425_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_1426_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_1427_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_1428_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( 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_1429_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( 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 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_1430_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_1431_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A3 @ B2 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_1432_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X3: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W @ Z2 ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_1433_vebt__member_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X3 @ Xa2 )
= Y )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y
= ( ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> Y )
=> ( ( ? [V: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> Y )
=> ( ( ? [V: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> Y )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary3: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
=> ( Y
= ( ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_1434_vebt__member_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ! [V: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ! [V: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary3: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_1435_invar__vebt_Ocases,axiom,
! [A1: vEBT_VEBT,A22: nat] :
( ( vEBT_invar_vebt @ A1 @ A22 )
=> ( ( ? [A5: $o,B4: $o] :
( A1
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( A22
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary3 @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N2 )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_1 )
=> ~ ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary3 @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_1 )
=> ~ ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary3 @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N2 )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X: nat] :
( ( ( ( vEBT_VEBT_high @ X @ N2 )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X )
& ( ord_less_eq @ nat @ X @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary3 @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X: nat] :
( ( ( ( vEBT_VEBT_high @ X @ N2 )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X )
& ( ord_less_eq @ nat @ X @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_1436_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A12: vEBT_VEBT,A23: nat] :
( ( ? [A8: $o,B8: $o] :
( A12
= ( vEBT_Leaf @ A8 @ B8 ) )
& ( A23
= ( suc @ ( zero_zero @ nat ) ) ) )
| ? [TreeList4: list @ vEBT_VEBT,N3: nat,Summary4: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A23 @ TreeList4 @ Summary4 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ( vEBT_invar_vebt @ X4 @ N3 ) )
& ( vEBT_invar_vebt @ Summary4 @ N3 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList4 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
& ( A23
= ( plus_plus @ nat @ N3 @ N3 ) )
& ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
| ? [TreeList4: list @ vEBT_VEBT,N3: nat,Summary4: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A23 @ TreeList4 @ Summary4 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ( vEBT_invar_vebt @ X4 @ N3 ) )
& ( vEBT_invar_vebt @ Summary4 @ ( suc @ N3 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList4 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
& ( A23
= ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) )
& ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
| ? [TreeList4: list @ vEBT_VEBT,N3: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList4 @ Summary4 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ( vEBT_invar_vebt @ X4 @ N3 ) )
& ( vEBT_invar_vebt @ Summary4 @ N3 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList4 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
& ( A23
= ( plus_plus @ nat @ N3 @ N3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary4 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A23 ) )
& ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList4: list @ vEBT_VEBT,N3: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList4 @ Summary4 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ( vEBT_invar_vebt @ X4 @ N3 ) )
& ( vEBT_invar_vebt @ Summary4 @ ( suc @ N3 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList4 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
& ( A23
= ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary4 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A23 ) )
& ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_1437_vebt__insert_Oelims,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X3 @ Xa2 )
= Y )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( vEBT_Leaf @ $true @ B4 ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A5 @ $true ) ) )
& ( ( Xa2
!= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) ) )
=> ( ! [V: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary3 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary3 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
=> ( Y
!= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary3 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
thf(fact_1438_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V2: A,R2: A,S: A] :
( ( ord_less_eq @ A @ U @ V2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 )
=> ( ( ord_less_eq @ A @ R2 @ S )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R2 @ ( minus_minus @ A @ V2 @ U ) ) @ S ) ) @ V2 ) ) ) ) ) ).
% scaling_mono
thf(fact_1439_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_1440_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_1441_vebt__succ_Opelims,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: option @ nat] :
( ( ( vEBT_vebt_succ @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [Uu: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y
= ( none @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv: $o,Uw: $o] :
( ( X3
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ! [N2: nat] :
( ( Xa2
= ( suc @ N2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
=> ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
=> ( ( ( ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( Y
= ( some @ nat @ Mi2 ) ) )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.pelims
thf(fact_1442_vebt__pred_Opelims,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: option @ nat] :
( ( ( vEBT_vebt_pred @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A5: $o,Uw: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ Uw ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( ( A5
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y
= ( none @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ! [Va3: nat] :
( ( Xa2
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A5
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
=> ( ( ( ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( Y
= ( some @ nat @ Ma2 ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi2 @ Xa2 ) @ ( some @ nat @ Mi2 ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.pelims
thf(fact_1443_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_1444_vebt__delete_Opelims,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ B4 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( vEBT_Leaf @ A5 @ $false ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ! [N2: nat] :
( ( Xa2
= ( suc @ ( suc @ N2 ) ) )
=> ( ( Y
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary3 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
=> ( ( ( ( ( ord_less @ nat @ Xa2 @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa2 ) )
=> ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) ) )
& ( ~ ( ( ord_less @ nat @ Xa2 @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa2 ) )
=> ( ( ( ( Xa2 = Mi2 )
& ( Xa2 = Ma2 ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) ) )
& ( ~ ( ( Xa2 = Mi2 )
& ( Xa2 = Ma2 ) )
=> ( Y
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa2 = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa2 != Mi2 )
=> ( Xa2 = Ma2 ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( Xa2 = Mi2 ) @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va3 ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa2 = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa2 != Mi2 )
=> ( Xa2 = Ma2 ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va3 ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary3 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_delete.pelims
thf(fact_1445_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_1446_Suc__if__eq,axiom,
! [A: $tType,F3: nat > A,H: nat > A,G3: A,N: nat] :
( ! [N2: nat] :
( ( F3 @ ( suc @ N2 ) )
= ( H @ N2 ) )
=> ( ( ( F3 @ ( zero_zero @ nat ) )
= G3 )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( ( F3 @ N )
= G3 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( F3 @ N )
= ( H @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% Suc_if_eq
thf(fact_1447_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_eq @ nat @ M5 @ N )
& ( P @ M5 ) ) )
= ( ? [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_1448_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_eq @ nat @ M5 @ N )
=> ( P @ M5 ) ) )
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_1449_vebt__insert_Opelims,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( vEBT_Leaf @ $true @ B4 ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A5 @ $true ) ) )
& ( ( Xa2
!= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) @ Xa2 ) ) ) )
=> ( ! [V: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary3 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V ) ) @ 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 @ V ) ) @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
=> ( ( Y
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary3 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ 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 @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
thf(fact_1450_insert_H__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_set_vebt @ ( vEBT_VEBT_insert @ T2 @ X3 ) )
= ( inf_inf @ ( set @ nat ) @ ( sup_sup @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert2 @ nat @ X3 @ ( bot_bot @ ( set @ nat ) ) ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% insert'_correct
thf(fact_1451_VEBT__internal_Oinsert_H_Opelims,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_VEBT_insert @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y
= ( vEBT_vebt_insert @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) @ Xa2 )
=> ( Y
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) ) )
& ( ~ ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) @ Xa2 )
=> ( Y
= ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.insert'.pelims
thf(fact_1452_vebt__member_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ Xa2 ) ) )
=> ( ! [V: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa2 ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ 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 @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ Xa2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_1453_vebt__member_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y
= ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
=> ( ( Y
= ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_1454_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y
= ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) ) ) )
=> ( ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Xa2 ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_1455_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) @ Xa2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_1456_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Xa2 ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) @ Xa2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_1457_IntI,axiom,
! [A: $tType,C3: A,A6: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ A6 )
=> ( ( member @ A @ C3 @ B5 )
=> ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ) ) ).
% IntI
thf(fact_1458_Int__iff,axiom,
! [A: $tType,C3: A,A6: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) )
= ( ( member @ A @ C3 @ A6 )
& ( member @ A @ C3 @ B5 ) ) ) ).
% Int_iff
thf(fact_1459_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C3 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% inf.bounded_iff
thf(fact_1460_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) )
= ( ( ord_less_eq @ A @ X3 @ Y )
& ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% le_inf_iff
thf(fact_1461_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ X3 @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_1462_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X3 )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_1463_Int__subset__iff,axiom,
! [A: $tType,C4: set @ A,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C4 @ A6 )
& ( ord_less_eq @ ( set @ A ) @ C4 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_1464_Int__insert__right__if1,axiom,
! [A: $tType,A3: A,A6: set @ A,B5: set @ A] :
( ( member @ A @ A3 @ A6 )
=> ( ( inf_inf @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ B5 ) )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1465_Int__insert__right__if0,axiom,
! [A: $tType,A3: A,A6: set @ A,B5: set @ A] :
( ~ ( member @ A @ A3 @ A6 )
=> ( ( inf_inf @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_1466_insert__inter__insert,axiom,
! [A: $tType,A3: A,A6: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ A6 ) @ ( insert2 @ A @ A3 @ B5 ) )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ) ).
% insert_inter_insert
thf(fact_1467_Int__insert__left__if1,axiom,
! [A: $tType,A3: A,C4: set @ A,B5: set @ A] :
( ( member @ A @ A3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B5 ) @ C4 )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1468_Int__insert__left__if0,axiom,
! [A: $tType,A3: A,C4: set @ A,B5: set @ A] :
( ~ ( member @ A @ A3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B5 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_1469_Int__Un__eq_I4_J,axiom,
! [A: $tType,T4: set @ A,S3: set @ A] :
( ( sup_sup @ ( set @ A ) @ T4 @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) )
= T4 ) ).
% Int_Un_eq(4)
thf(fact_1470_Int__Un__eq_I3_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ S3 @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_1471_Int__Un__eq_I2_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) @ T4 )
= T4 ) ).
% Int_Un_eq(2)
thf(fact_1472_Int__Un__eq_I1_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_1473_Un__Int__eq_I4_J,axiom,
! [A: $tType,T4: set @ A,S3: set @ A] :
( ( inf_inf @ ( set @ A ) @ T4 @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) )
= T4 ) ).
% Un_Int_eq(4)
thf(fact_1474_Un__Int__eq_I3_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ S3 @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_1475_Un__Int__eq_I2_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) @ T4 )
= T4 ) ).
% Un_Int_eq(2)
thf(fact_1476_Un__Int__eq_I1_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_1477_insert__disjoint_I1_J,axiom,
! [A: $tType,A3: A,A6: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ A6 ) @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A3 @ B5 )
& ( ( inf_inf @ ( set @ A ) @ A6 @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_1478_insert__disjoint_I2_J,axiom,
! [A: $tType,A3: A,A6: set @ A,B5: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ A6 ) @ B5 ) )
= ( ~ ( member @ A @ A3 @ B5 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1479_disjoint__insert_I1_J,axiom,
! [A: $tType,B5: set @ A,A3: A,A6: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B5 @ ( insert2 @ A @ A3 @ A6 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A3 @ B5 )
& ( ( inf_inf @ ( set @ A ) @ B5 @ A6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_1480_disjoint__insert_I2_J,axiom,
! [A: $tType,A6: set @ A,B2: A,B5: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A6 @ ( insert2 @ A @ B2 @ B5 ) ) )
= ( ~ ( member @ A @ B2 @ A6 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1481_Diff__disjoint,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ ( minus_minus @ ( set @ A ) @ B5 @ A6 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_1482_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_1483_Int__Collect,axiom,
! [A: $tType,X3: A,A6: set @ A,P: A > $o] :
( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A6 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X3 @ A6 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_1484_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A7 )
& ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_1485_IntE,axiom,
! [A: $tType,C3: A,A6: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) )
=> ~ ( ( member @ A @ C3 @ A6 )
=> ~ ( member @ A @ C3 @ B5 ) ) ) ).
% IntE
thf(fact_1486_IntD1,axiom,
! [A: $tType,C3: A,A6: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) )
=> ( member @ A @ C3 @ A6 ) ) ).
% IntD1
thf(fact_1487_IntD2,axiom,
! [A: $tType,C3: A,A6: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) )
=> ( member @ A @ C3 @ B5 ) ) ).
% IntD2
thf(fact_1488_Int__assoc,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ A6 @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Int_assoc
thf(fact_1489_Int__absorb,axiom,
! [A: $tType,A6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ A6 )
= A6 ) ).
% Int_absorb
thf(fact_1490_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ B6 @ A7 ) ) ) ).
% Int_commute
thf(fact_1491_Int__left__absorb,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ).
% Int_left_absorb
thf(fact_1492_Int__left__commute,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) )
= ( inf_inf @ ( set @ A ) @ B5 @ ( inf_inf @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% Int_left_commute
thf(fact_1493_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% inf.coboundedI2
thf(fact_1494_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% inf.coboundedI1
thf(fact_1495_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B8: A,A8: A] :
( ( inf_inf @ A @ A8 @ B8 )
= B8 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_1496_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] :
( ( inf_inf @ A @ A8 @ B8 )
= A8 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_1497_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 ) ) ).
% inf.cobounded2
thf(fact_1498_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ A3 ) ) ).
% inf.cobounded1
thf(fact_1499_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] :
( A8
= ( inf_inf @ A @ A8 @ B8 ) ) ) ) ) ).
% inf.order_iff
thf(fact_1500_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ) ).
% inf_greatest
thf(fact_1501_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C3 ) ) ) ) ) ).
% inf.boundedI
thf(fact_1502_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C3 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% inf.boundedE
thf(fact_1503_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( inf_inf @ A @ X3 @ Y )
= Y ) ) ) ).
% inf_absorb2
thf(fact_1504_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( inf_inf @ A @ X3 @ Y )
= X3 ) ) ) ).
% inf_absorb1
thf(fact_1505_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb2
thf(fact_1506_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb1
thf(fact_1507_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y3: A] :
( ( inf_inf @ A @ X4 @ Y3 )
= X4 ) ) ) ) ).
% le_iff_inf
thf(fact_1508_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F3: A > A > A,X3: A,Y: A] :
( ! [X5: A,Y4: A] : ( ord_less_eq @ A @ ( F3 @ X5 @ Y4 ) @ X5 )
=> ( ! [X5: A,Y4: A] : ( ord_less_eq @ A @ ( F3 @ X5 @ Y4 ) @ Y4 )
=> ( ! [X5: A,Y4: A,Z3: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ( ord_less_eq @ A @ X5 @ Z3 )
=> ( ord_less_eq @ A @ X5 @ ( F3 @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf @ A @ X3 @ Y )
= ( F3 @ X3 @ Y ) ) ) ) ) ) ).
% inf_unique
thf(fact_1509_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( inf_inf @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% inf.orderI
thf(fact_1510_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3
= ( inf_inf @ A @ A3 @ B2 ) ) ) ) ).
% inf.orderE
thf(fact_1511_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X3: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ X3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X3 ) ) ) ).
% le_infI2
thf(fact_1512_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ X3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X3 ) ) ) ).
% le_infI1
thf(fact_1513_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C3: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ ( inf_inf @ A @ C3 @ D3 ) ) ) ) ) ).
% inf_mono
thf(fact_1514_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X3 @ A3 )
=> ( ( ord_less_eq @ A @ X3 @ B2 )
=> ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% le_infI
thf(fact_1515_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A3 @ B2 ) )
=> ~ ( ( ord_less_eq @ A @ X3 @ A3 )
=> ~ ( ord_less_eq @ A @ X3 @ B2 ) ) ) ) ).
% le_infE
thf(fact_1516_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ Y ) ) ).
% inf_le2
thf(fact_1517_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ X3 ) ) ).
% inf_le1
thf(fact_1518_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ X3 ) ) ).
% inf_sup_ord(1)
thf(fact_1519_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ Y ) ) ).
% inf_sup_ord(2)
thf(fact_1520_disjoint__iff__not__equal,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A6 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ B5 )
=> ( X4 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1521_Int__empty__right,axiom,
! [A: $tType,A6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_1522_Int__empty__left,axiom,
! [A: $tType,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_1523_disjoint__iff,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A6 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ~ ( member @ A @ X4 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_1524_Int__emptyI,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ~ ( member @ A @ X5 @ B5 ) )
=> ( ( inf_inf @ ( set @ A ) @ A6 @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_1525_Int__Collect__mono,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B5 @ ( collect @ A @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1526_Int__greatest,axiom,
! [A: $tType,C4: set @ A,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ C4 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_1527_Int__absorb2,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( inf_inf @ ( set @ A ) @ A6 @ B5 )
= A6 ) ) ).
% Int_absorb2
thf(fact_1528_Int__absorb1,axiom,
! [A: $tType,B5: set @ A,A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ( inf_inf @ ( set @ A ) @ A6 @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_1529_Int__lower2,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_1530_Int__lower1,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) @ A6 ) ).
% Int_lower1
thf(fact_1531_Int__mono,axiom,
! [A: $tType,A6: set @ A,C4: set @ A,B5: set @ A,D4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) @ ( inf_inf @ ( set @ A ) @ C4 @ D4 ) ) ) ) ).
% Int_mono
thf(fact_1532_Int__insert__right,axiom,
! [A: $tType,A3: A,A6: set @ A,B5: set @ A] :
( ( ( member @ A @ A3 @ A6 )
=> ( ( inf_inf @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ B5 ) )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ) )
& ( ~ ( member @ A @ A3 @ A6 )
=> ( ( inf_inf @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_1533_Int__insert__left,axiom,
! [A: $tType,A3: A,C4: set @ A,B5: set @ A] :
( ( ( member @ A @ A3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B5 ) @ C4 )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) ) ) )
& ( ~ ( member @ A @ A3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B5 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ B5 @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_1534_Un__Int__crazy,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) ) @ ( inf_inf @ ( set @ A ) @ C4 @ A6 ) )
= ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) ) @ ( sup_sup @ ( set @ A ) @ C4 @ A6 ) ) ) ).
% Un_Int_crazy
thf(fact_1535_Int__Un__distrib,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) @ ( inf_inf @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% Int_Un_distrib
thf(fact_1536_Un__Int__distrib,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) @ ( sup_sup @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% Un_Int_distrib
thf(fact_1537_Int__Un__distrib2,axiom,
! [A: $tType,B5: set @ A,C4: set @ A,A6: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) @ A6 )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B5 @ A6 ) @ ( inf_inf @ ( set @ A ) @ C4 @ A6 ) ) ) ).
% Int_Un_distrib2
thf(fact_1538_Un__Int__distrib2,axiom,
! [A: $tType,B5: set @ A,C4: set @ A,A6: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) @ A6 )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B5 @ A6 ) @ ( sup_sup @ ( set @ A ) @ C4 @ A6 ) ) ) ).
% Un_Int_distrib2
thf(fact_1539_Int__Diff,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ A6 @ ( minus_minus @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Int_Diff
thf(fact_1540_Diff__Int2,axiom,
! [A: $tType,A6: set @ A,C4: set @ A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ C4 ) @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ C4 ) @ B5 ) ) ).
% Diff_Int2
thf(fact_1541_Diff__Diff__Int,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ).
% Diff_Diff_Int
thf(fact_1542_Diff__Int__distrib,axiom,
! [A: $tType,C4: set @ A,A6: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ C4 @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C4 @ A6 ) @ ( inf_inf @ ( set @ A ) @ C4 @ B5 ) ) ) ).
% Diff_Int_distrib
thf(fact_1543_Diff__Int__distrib2,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) @ C4 )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ C4 ) @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1544_distrib__sup__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y ) @ ( sup_sup @ A @ X3 @ Z2 ) ) ) ) ).
% distrib_sup_le
thf(fact_1545_distrib__inf__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y ) @ ( inf_inf @ A @ X3 @ Z2 ) ) @ ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% distrib_inf_le
thf(fact_1546_Diff__triv,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A6 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( minus_minus @ ( set @ A ) @ A6 @ B5 )
= A6 ) ) ).
% Diff_triv
thf(fact_1547_Int__Diff__disjoint,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_Diff_disjoint
thf(fact_1548_Un__Int__assoc__eq,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) ) )
= ( ord_less_eq @ ( set @ A ) @ C4 @ A6 ) ) ).
% Un_Int_assoc_eq
thf(fact_1549_Diff__Un,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) )
= ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) @ ( minus_minus @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% Diff_Un
thf(fact_1550_Diff__Int,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) @ ( minus_minus @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% Diff_Int
thf(fact_1551_Int__Diff__Un,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) )
= A6 ) ).
% Int_Diff_Un
thf(fact_1552_Un__Diff__Int,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) )
= A6 ) ).
% Un_Diff_Int
thf(fact_1553_vebt__member_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ 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 @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ Xa2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_1554_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ( ( Y
= ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
=> ( ( Y
= ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [V: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd2 ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( 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 @ V ) @ TreeList2 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_1555_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd2 ) @ Xa2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_1556_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd2 ) @ Xa2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_1557_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( insert2 @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B2 )
& ( B2 = C3 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_1558_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_1559_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_1560_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 )
| ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_1561_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_1562_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_1563_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_1564_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_1565_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R )
@ ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ S3 ) )
= ( ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ) ) ).
% inf_Int_eq2
thf(fact_1566_inf__Int__eq,axiom,
! [A: $tType,R: set @ A,S3: set @ A] :
( ( inf_inf @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R )
@ ^ [X4: A] : ( member @ A @ X4 @ S3 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( inf_inf @ ( set @ A ) @ R @ S3 ) ) ) ) ).
% inf_Int_eq
thf(fact_1567_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_1568_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M7: nat] :
( ( P @ X3 )
=> ( ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq @ nat @ X5 @ M7 ) )
=> ~ ! [M: nat] :
( ( P @ M )
=> ~ ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq @ nat @ X @ M ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1569_fold__atLeastAtMost__nat_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
~ ! [F2: nat > A > A,A5: nat,B4: nat,Acc: A] :
( X3
!= ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A5 @ ( product_Pair @ nat @ A @ B4 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_1570_ivl__disj__un__two__touch_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_1571_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_1572_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( ( ~ ( ord_less_eq @ A @ A3 @ B2 )
| ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 )
& ( ( ord_less @ A @ C3 @ A3 )
| ( ord_less @ A @ B2 @ D3 ) ) ) )
& ( ord_less_eq @ A @ C3 @ D3 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1573_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_1574_Icc__eq__insert__lb__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or1337092689740270186AtMost @ nat @ M2 @ N )
= ( insert2 @ nat @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_1575_atLeastAtMostSuc__conv,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) )
= ( insert2 @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_1576_atLeastAtMost__insertL,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( insert2 @ nat @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_1577_boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ X3 @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_right
thf(fact_1578_boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X3 )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_left
thf(fact_1579_set__union,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( union @ A @ Xs2 @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ Ys ) ) ) ).
% set_union
thf(fact_1580_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ Y @ Z2 ) )
= ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1581_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ X3 ) @ ( times_times @ A @ Z2 @ Y ) )
= ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1582_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q3: A,R2: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q3 @ R2 ) )
= ( R2
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_1583_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X4: nat,N3: nat] : ( modulo_modulo @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% low_def
thf(fact_1584_triangle__def,axiom,
( nat_triangle
= ( ^ [N3: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% triangle_def
thf(fact_1585_mod__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_add_self2
thf(fact_1586_mod__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_add_self1
thf(fact_1587_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_1588_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C3 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self1
thf(fact_1589_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self2
thf(fact_1590_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C3 @ B2 ) @ A3 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self3
thf(fact_1591_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C3 ) @ A3 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self4
thf(fact_1592_mod__by__Suc__0,axiom,
! [M2: nat] :
( ( modulo_modulo @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_1593_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_1594_Suc__mod__mult__self4,axiom,
! [N: nat,K2: nat,M2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K2 ) @ M2 ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_1595_Suc__mod__mult__self3,axiom,
! [K2: nat,N: nat,M2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ N ) @ M2 ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_1596_Suc__mod__mult__self2,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M2 @ ( times_times @ nat @ N @ K2 ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_1597_Suc__mod__mult__self1,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M2 @ ( times_times @ nat @ K2 @ N ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_1598_one__mod__two__eq__one,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% one_mod_two_eq_one
thf(fact_1599_bits__one__mod__two__eq__one,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% bits_one_mod_two_eq_one
thf(fact_1600_mod2__Suc__Suc,axiom,
! [M2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ M2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% mod2_Suc_Suc
thf(fact_1601_Suc__times__numeral__mod__eq,axiom,
! [K2: num,N: nat] :
( ( ( numeral_numeral @ nat @ K2 )
!= ( one_one @ nat ) )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K2 ) @ N ) ) @ ( numeral_numeral @ nat @ K2 ) )
= ( one_one @ nat ) ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_1602_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_1603_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_1604_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_1605_add__self__mod__2,axiom,
! [M2: nat] :
( ( modulo_modulo @ nat @ ( plus_plus @ nat @ M2 @ M2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ nat ) ) ).
% add_self_mod_2
thf(fact_1606_mod2__gr__0,axiom,
! [M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ nat ) ) ) ).
% mod2_gr_0
thf(fact_1607_mod__add__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C3 ) ) @ C3 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% mod_add_right_eq
thf(fact_1608_mod__add__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C3 ) @ B2 ) @ C3 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% mod_add_left_eq
thf(fact_1609_mod__add__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,A4: A,B2: A,B3: A] :
( ( ( modulo_modulo @ A @ A3 @ C3 )
= ( modulo_modulo @ A @ A4 @ C3 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C3 )
= ( modulo_modulo @ A @ B3 @ C3 ) )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C3 ) ) ) ) ) ).
% mod_add_cong
thf(fact_1610_mod__add__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C3 ) @ ( modulo_modulo @ A @ B2 @ C3 ) ) @ C3 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% mod_add_eq
thf(fact_1611_power__mod,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ N ) @ B2 )
= ( modulo_modulo @ A @ ( power_power @ A @ A3 @ N ) @ B2 ) ) ) ).
% power_mod
thf(fact_1612_mod__Suc__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( modulo_modulo @ nat @ M2 @ N ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ ( suc @ M2 ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_1613_mod__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( modulo_modulo @ nat @ M2 @ N ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% mod_Suc_eq
thf(fact_1614_mod__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M2 @ N ) @ M2 ) ).
% mod_less_eq_dividend
thf(fact_1615_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ A3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_1616_cong__exp__iff__simps_I9_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,Q3: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ Q3 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_1617_cong__exp__iff__simps_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ one2 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_1618_mod__eqE,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ C3 )
= ( modulo_modulo @ A @ B2 @ C3 ) )
=> ~ ! [D2: A] :
( B2
!= ( plus_plus @ A @ A3 @ ( times_times @ A @ C3 @ D2 ) ) ) ) ) ).
% mod_eqE
thf(fact_1619_div__add1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C3 ) @ ( modulo_modulo @ A @ B2 @ C3 ) ) @ C3 ) ) ) ) ).
% div_add1_eq
thf(fact_1620_mod__Suc,axiom,
! [M2: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo @ nat @ M2 @ N ) )
= N )
=> ( ( modulo_modulo @ nat @ ( suc @ M2 ) @ N )
= ( zero_zero @ nat ) ) )
& ( ( ( suc @ ( modulo_modulo @ nat @ M2 @ N ) )
!= N )
=> ( ( modulo_modulo @ nat @ ( suc @ M2 ) @ N )
= ( suc @ ( modulo_modulo @ nat @ M2 @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_1621_mod__induct,axiom,
! [P: nat > $o,N: nat,P2: nat,M2: nat] :
( ( P @ N )
=> ( ( ord_less @ nat @ N @ P2 )
=> ( ( ord_less @ nat @ M2 @ P2 )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ P2 )
=> ( ( P @ N2 )
=> ( P @ ( modulo_modulo @ nat @ ( suc @ N2 ) @ P2 ) ) ) )
=> ( P @ M2 ) ) ) ) ) ).
% mod_induct
thf(fact_1622_mod__Suc__le__divisor,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M2 @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_1623_le__mod__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( modulo_modulo @ nat @ M2 @ N )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ).
% le_mod_geq
thf(fact_1624_nat__mod__eq__iff,axiom,
! [X3: nat,N: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X3 @ N )
= ( modulo_modulo @ nat @ Y @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus @ nat @ X3 @ ( times_times @ nat @ N @ Q1 ) )
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_1625_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_1626_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= A3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_1627_cong__exp__iff__simps_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q3: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q3 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(2)
thf(fact_1628_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_1629_cong__exp__iff__simps_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q3: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_1630_cong__exp__iff__simps_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,Q3: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_1631_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C3 )
= ( plus_plus @ A @ A3 @ C3 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_1632_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C3 )
= ( plus_plus @ A @ A3 @ C3 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_1633_mod__div__decomp,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( A3
= ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% mod_div_decomp
thf(fact_1634_div__mult__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
= A3 ) ) ).
% div_mult_mod_eq
thf(fact_1635_mod__div__mult__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
= A3 ) ) ).
% mod_div_mult_eq
thf(fact_1636_mod__mult__div__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
= A3 ) ) ).
% mod_mult_div_eq
thf(fact_1637_mult__div__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [B2: A,A3: A] :
( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
= A3 ) ) ).
% mult_div_mod_eq
thf(fact_1638_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C3 ) ) @ C3 ) ) ) ) ).
% div_mult1_eq
thf(fact_1639_mod__le__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M2 @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_1640_nat__mod__eq__lemma,axiom,
! [X3: nat,N: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X3 @ N )
= ( modulo_modulo @ nat @ Y @ N ) )
=> ( ( ord_less_eq @ nat @ Y @ X3 )
=> ? [Q2: nat] :
( X3
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q2 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_1641_mod__eq__nat2E,axiom,
! [M2: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M2 @ Q3 )
= ( modulo_modulo @ nat @ N @ Q3 ) )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ~ ! [S2: nat] :
( N
!= ( plus_plus @ nat @ M2 @ ( times_times @ nat @ Q3 @ S2 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_1642_mod__eq__nat1E,axiom,
! [M2: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M2 @ Q3 )
= ( modulo_modulo @ nat @ N @ Q3 ) )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ~ ! [S2: nat] :
( M2
!= ( plus_plus @ nat @ N @ ( times_times @ nat @ Q3 @ S2 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_1643_mod__mult2__eq,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( modulo_modulo @ nat @ M2 @ ( times_times @ nat @ N @ Q3 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ N @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M2 @ N ) @ Q3 ) ) @ ( modulo_modulo @ nat @ M2 @ N ) ) ) ).
% mod_mult2_eq
thf(fact_1644_split__mod,axiom,
! [P: nat > $o,M2: nat,N: nat] :
( ( P @ ( modulo_modulo @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ M2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I4 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_1645_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_1646_Suc__times__mod__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M2 @ N ) ) @ M2 )
= ( one_one @ nat ) ) ) ).
% Suc_times_mod_eq
thf(fact_1647_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_1648_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_1649_div__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat,M2: nat] :
( ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M2 ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_1650_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_1651_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_1652_boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ ( bot_bot @ A ) )
= X3 ) ) ).
% boolean_algebra.disj_zero_right
thf(fact_1653_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: A,X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ( modulo_modulo @ A @ X3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( modulo_modulo @ A @ X3 @ M2 ) )
| ( ( modulo_modulo @ A @ X3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X3 @ M2 ) @ M2 ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_1654_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_1655_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_1656_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_1657_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_1658_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ( minus_minus @ A @ X3 @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% diff_shunt_var
thf(fact_1659_verit__le__mono__div,axiom,
! [A6: nat,B5: nat,N: nat] :
( ( ord_less @ nat @ A6 @ B5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A6 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B5 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B5 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_1660_div__mod__decomp,axiom,
! [A6: nat,N: nat] :
( A6
= ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A6 @ N ) @ N ) @ ( modulo_modulo @ nat @ A6 @ N ) ) ) ).
% div_mod_decomp
thf(fact_1661_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_1662_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs2: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs2 @ Ys ) @ N )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1663_obtain__set__succ,axiom,
! [X3: nat,Z2: nat,A6: set @ nat,B5: set @ nat] :
( ( ord_less @ nat @ X3 @ Z2 )
=> ( ( vEBT_VEBT_max_in_set @ A6 @ Z2 )
=> ( ( finite_finite2 @ nat @ B5 )
=> ( ( A6 = B5 )
=> ? [X_12: nat] : ( vEBT_is_succ_in_set @ A6 @ X3 @ X_12 ) ) ) ) ) ).
% obtain_set_succ
thf(fact_1664_obtain__set__pred,axiom,
! [Z2: nat,X3: nat,A6: set @ nat] :
( ( ord_less @ nat @ Z2 @ X3 )
=> ( ( vEBT_VEBT_min_in_set @ A6 @ Z2 )
=> ( ( finite_finite2 @ nat @ A6 )
=> ? [X_12: nat] : ( vEBT_is_pred_in_set @ A6 @ X3 @ X_12 ) ) ) ) ).
% obtain_set_pred
thf(fact_1665_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q3: int,R2: int] :
( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A3 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q3 @ 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 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q3 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_1666_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q3: int,R2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q3 @ 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 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q3 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_1667_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_1668_pred__none__empty,axiom,
! [Xs2: set @ nat,A3: nat] :
( ~ ? [X_12: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A3 @ X_12 )
=> ( ( finite_finite2 @ nat @ Xs2 )
=> ~ ? [X: nat] :
( ( member @ nat @ X @ Xs2 )
& ( ord_less @ nat @ X @ A3 ) ) ) ) ).
% pred_none_empty
thf(fact_1669_succ__none__empty,axiom,
! [Xs2: set @ nat,A3: nat] :
( ~ ? [X_12: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A3 @ X_12 )
=> ( ( finite_finite2 @ nat @ Xs2 )
=> ~ ? [X: nat] :
( ( member @ nat @ X @ Xs2 )
& ( ord_less @ nat @ A3 @ X ) ) ) ) ).
% succ_none_empty
thf(fact_1670_verit__eq__simplify_I8_J,axiom,
! [X2: num,Y2: num] :
( ( ( bit0 @ X2 )
= ( bit0 @ Y2 ) )
= ( X2 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_1671_List_Ofinite__set,axiom,
! [A: $tType,Xs2: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs2 ) ) ).
% List.finite_set
thf(fact_1672_length__product,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs2 @ Ys ) )
= ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% length_product
thf(fact_1673_zmod__numeral__Bit0,axiom,
! [V2: num,W: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit0 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_1674_eucl__rel__int,axiom,
! [K2: int,L: int] : ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K2 @ L ) @ ( modulo_modulo @ int @ K2 @ L ) ) ) ).
% eucl_rel_int
thf(fact_1675_mod__int__unique,axiom,
! [K2: int,L: int,Q3: int,R2: int] :
( ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( modulo_modulo @ int @ K2 @ L )
= R2 ) ) ).
% mod_int_unique
thf(fact_1676_unique__quotient,axiom,
! [A3: int,B2: int,Q3: int,R2: int,Q5: int,R4: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q5 @ R4 ) )
=> ( Q3 = Q5 ) ) ) ).
% unique_quotient
thf(fact_1677_unique__remainder,axiom,
! [A3: int,B2: int,Q3: int,R2: int,Q5: int,R4: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q5 @ R4 ) )
=> ( R2 = R4 ) ) ) ).
% unique_remainder
thf(fact_1678_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N6: set @ nat] :
? [M5: nat] :
! [X4: nat] :
( ( member @ nat @ X4 @ N6 )
=> ( ord_less_eq @ nat @ X4 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1679_finite__list,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ? [Xs3: list @ A] :
( ( set2 @ A @ Xs3 )
= A6 ) ) ).
% finite_list
thf(fact_1680_finite__less__ub,axiom,
! [F3: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( F3 @ N2 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ ( F3 @ N3 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1681_finite__lists__length__eq,axiom,
! [A: $tType,A6: set @ A,N: nat] :
( ( finite_finite2 @ A @ A6 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A6 )
& ( ( size_size @ ( list @ A ) @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1682_eucl__rel__int__by0,axiom,
! [K2: int] : ( eucl_rel_int @ K2 @ ( zero_zero @ int ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K2 ) ) ).
% eucl_rel_int_by0
thf(fact_1683_div__int__unique,axiom,
! [K2: int,L: int,Q3: int,R2: int] :
( ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( divide_divide @ int @ K2 @ L )
= Q3 ) ) ).
% div_int_unique
thf(fact_1684_finite__lists__length__le,axiom,
! [A: $tType,A6: set @ A,N: nat] :
( ( finite_finite2 @ A @ A6 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A6 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1685_eucl__rel__int__dividesI,axiom,
! [L: int,K2: int,Q3: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( K2
= ( times_times @ int @ Q3 @ L ) )
=> ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_1686_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(2)
thf(fact_1687_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
| ~ ( ord_less_eq @ A @ A3 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% verit_la_disequality
thf(fact_1688_subset__eq__atLeast0__atMost__finite,axiom,
! [N5: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N5 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N5 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_1689_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B3: B,A4: B] :
( ( ~ ( ord_less_eq @ B @ B3 @ A4 ) )
= ( ord_less @ B @ A4 @ B3 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_1690_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% verit_sum_simplify
thf(fact_1691_verit__eq__simplify_I10_J,axiom,
! [X2: num] :
( one2
!= ( bit0 @ X2 ) ) ).
% verit_eq_simplify(10)
thf(fact_1692_eucl__rel__int__iff,axiom,
! [K2: int,L: int,Q3: int,R2: int] :
( ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
= ( ( K2
= ( plus_plus @ int @ ( times_times @ int @ L @ Q3 ) @ 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 ) )
=> ( Q3
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_1693_pos__zmod__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ A3 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_1694_neg__zmod__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) @ ( one_one @ int ) ) ) ) ).
% neg_zmod_mult_2
thf(fact_1695_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A8: A,B8: A] : ( if @ A @ ( ord_less_eq @ A @ A8 @ B8 ) @ B8 @ A8 ) ) ) ) ).
% max_def_raw
thf(fact_1696_finite__Diff__insert,axiom,
! [A: $tType,A6: set @ A,A3: A,B5: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ B5 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) ) ) ).
% finite_Diff_insert
thf(fact_1697_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_1698_finite__Collect__subsets,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1699_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
@ ^ [Z4: A] :
( ( power_power @ A @ Z4 @ N )
= ( one_one @ A ) ) ) ) ) ) ).
% finite_roots_unity
thf(fact_1700_finite__induct__select,axiom,
! [A: $tType,S3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T5: set @ A] :
( ( ord_less @ ( set @ A ) @ T5 @ S3 )
=> ( ( P @ T5 )
=> ? [X: A] :
( ( member @ A @ X @ ( minus_minus @ ( set @ A ) @ S3 @ T5 ) )
& ( P @ ( insert2 @ A @ X @ T5 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ).
% finite_induct_select
thf(fact_1701_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B5: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B5 )
=> ( P @ B5 ) )
=> ( ! [A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ( A10
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A10 @ B5 )
=> ( ! [X: A] :
( ( member @ A @ X @ A10 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% remove_induct
thf(fact_1702_finite__remove__induct,axiom,
! [A: $tType,B5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ( A10
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A10 @ B5 )
=> ( ! [X: A] :
( ( member @ A @ X @ A10 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_1703_finite__insert,axiom,
! [A: $tType,A3: A,A6: set @ A] :
( ( finite_finite2 @ A @ ( insert2 @ A @ A3 @ A6 ) )
= ( finite_finite2 @ A @ A6 ) ) ).
% finite_insert
thf(fact_1704_finite__maxlen,axiom,
! [A: $tType,M7: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M7 )
=> ? [N2: nat] :
! [X: list @ A] :
( ( member @ ( list @ A ) @ X @ M7 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_1705_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A6 )
& ( ord_less_eq @ A @ A3 @ X5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A6 )
=> ( ( ord_less_eq @ A @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1706_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A6 )
& ( ord_less_eq @ A @ X5 @ A3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A6 )
=> ( ( ord_less_eq @ A @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1707_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_1708_infinite__imp__nonempty,axiom,
! [A: $tType,S3: set @ A] :
( ~ ( finite_finite2 @ A @ S3 )
=> ( S3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_1709_rev__finite__subset,axiom,
! [A: $tType,B5: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( finite_finite2 @ A @ A6 ) ) ) ).
% rev_finite_subset
thf(fact_1710_infinite__super,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ T4 )
=> ( ~ ( finite_finite2 @ A @ S3 )
=> ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% infinite_super
thf(fact_1711_finite__subset,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( finite_finite2 @ A @ A6 ) ) ) ).
% finite_subset
thf(fact_1712_finite_OinsertI,axiom,
! [A: $tType,A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( finite_finite2 @ A @ ( insert2 @ A @ A3 @ A6 ) ) ) ).
% finite.insertI
thf(fact_1713_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A6 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A6 )
=> ( ( ord_less_eq @ A @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1714_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A6 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A6 )
=> ( ( ord_less_eq @ A @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1715_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A6: set @ A] :
( ! [A10: set @ A] :
( ~ ( finite_finite2 @ A @ A10 )
=> ( P @ A10 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,F5: set @ A] :
( ( finite_finite2 @ A @ F5 )
=> ( ~ ( member @ A @ X5 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert2 @ A @ X5 @ F5 ) ) ) ) )
=> ( P @ A6 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1716_finite__ne__induct,axiom,
! [A: $tType,F6: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F6 )
=> ( ( F6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] : ( P @ ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X5: A,F5: set @ A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( F5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X5 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert2 @ A @ X5 @ F5 ) ) ) ) ) )
=> ( P @ F6 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1717_finite__induct,axiom,
! [A: $tType,F6: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F6 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,F5: set @ A] :
( ( finite_finite2 @ A @ F5 )
=> ( ~ ( member @ A @ X5 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert2 @ A @ X5 @ F5 ) ) ) ) )
=> ( P @ F6 ) ) ) ) ).
% finite_induct
thf(fact_1718_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A8: set @ A] :
( ( A8
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,B8: A] :
( ( A8
= ( insert2 @ A @ B8 @ A7 ) )
& ( finite_finite2 @ A @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_1719_finite_Ocases,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A10: set @ A] :
( ? [A5: A] :
( A3
= ( insert2 @ A @ A5 @ A10 ) )
=> ~ ( finite_finite2 @ A @ A10 ) ) ) ) ).
% finite.cases
thf(fact_1720_finite__subset__induct_H,axiom,
! [A: $tType,F6: set @ A,A6: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F6 )
=> ( ( ord_less_eq @ ( set @ A ) @ F6 @ A6 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A,F5: set @ A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( member @ A @ A5 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A6 )
=> ( ~ ( member @ A @ A5 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert2 @ A @ A5 @ F5 ) ) ) ) ) ) )
=> ( P @ F6 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1721_finite__subset__induct,axiom,
! [A: $tType,F6: set @ A,A6: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F6 )
=> ( ( ord_less_eq @ ( set @ A ) @ F6 @ A6 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A,F5: set @ A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( member @ A @ A5 @ A6 )
=> ( ~ ( member @ A @ A5 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert2 @ A @ A5 @ F5 ) ) ) ) ) )
=> ( P @ F6 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1722_finite__empty__induct,axiom,
! [A: $tType,A6: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A6 )
=> ( ( P @ A6 )
=> ( ! [A5: A,A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ( member @ A @ A5 @ A10 )
=> ( ( P @ A10 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_1723_infinite__coinduct,axiom,
! [A: $tType,X6: ( set @ A ) > $o,A6: set @ A] :
( ( X6 @ A6 )
=> ( ! [A10: set @ A] :
( ( X6 @ A10 )
=> ? [X: A] :
( ( member @ A @ X @ A10 )
& ( ( X6 @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A6 ) ) ) ).
% infinite_coinduct
thf(fact_1724_infinite__remove,axiom,
! [A: $tType,S3: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ S3 )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_1725_finite__nth__roots,axiom,
! [N: nat,C3: complex] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ complex
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= C3 ) ) ) ) ).
% finite_nth_roots
thf(fact_1726_set__encode__insert,axiom,
! [A6: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ A6 )
=> ( ~ ( member @ nat @ N @ A6 )
=> ( ( nat_set_encode @ ( insert2 @ nat @ N @ A6 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A6 ) ) ) ) ) ).
% set_encode_insert
thf(fact_1727_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A6 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B4: A,A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ! [X: A] :
( ( member @ A @ X @ A10 )
=> ( ord_less @ A @ B4 @ X ) )
=> ( ( P @ A10 )
=> ( P @ ( insert2 @ A @ B4 @ A10 ) ) ) ) )
=> ( P @ A6 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1728_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A6 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B4: A,A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ! [X: A] :
( ( member @ A @ X @ A10 )
=> ( ord_less @ A @ X @ B4 ) )
=> ( ( P @ A10 )
=> ( P @ ( insert2 @ A @ B4 @ A10 ) ) ) ) )
=> ( P @ A6 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1729_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S3: set @ B,P: ( set @ B ) > $o,F3: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X5: B,S4: set @ B] :
( ( finite_finite2 @ B @ S4 )
=> ( ! [Y6: B] :
( ( member @ B @ Y6 @ S4 )
=> ( ord_less_eq @ A @ ( F3 @ Y6 ) @ ( F3 @ X5 ) ) )
=> ( ( P @ S4 )
=> ( P @ ( insert2 @ B @ X5 @ S4 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_1730_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,X3: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I5 )
& ( ( X3 @ 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 @ ( X3 @ I4 ) @ ( Y @ I4 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_1731_concat__bit__Suc,axiom,
! [N: nat,K2: int,L: int] :
( ( bit_concat_bit @ ( suc @ N ) @ K2 @ L )
= ( plus_plus @ int @ ( modulo_modulo @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_1732_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_1733_concat__bit__assoc,axiom,
! [N: nat,K2: int,M2: nat,L: int,R2: int] :
( ( bit_concat_bit @ N @ K2 @ ( bit_concat_bit @ M2 @ L @ R2 ) )
= ( bit_concat_bit @ ( plus_plus @ nat @ M2 @ N ) @ ( bit_concat_bit @ N @ K2 @ L ) @ R2 ) ) ).
% concat_bit_assoc
thf(fact_1734_set__encode__eq,axiom,
! [A6: set @ nat,B5: set @ nat] :
( ( finite_finite2 @ nat @ A6 )
=> ( ( finite_finite2 @ nat @ B5 )
=> ( ( ( nat_set_encode @ A6 )
= ( nat_set_encode @ B5 ) )
= ( A6 = B5 ) ) ) ) ).
% set_encode_eq
thf(fact_1735_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K2: A,M2: A > nat] :
( ( P @ K2 )
=> ? [X5: A] :
( ( P @ X5 )
& ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ nat @ ( M2 @ X5 ) @ ( M2 @ Y6 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_1736_set__encode__inf,axiom,
! [A6: set @ nat] :
( ~ ( finite_finite2 @ nat @ A6 )
=> ( ( nat_set_encode @ A6 )
= ( zero_zero @ nat ) ) ) ).
% set_encode_inf
thf(fact_1737_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K2: A,F3: A > nat,B2: nat] :
( ( P @ K2 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less @ nat @ ( F3 @ Y4 ) @ B2 ) )
=> ? [X5: A] :
( ( P @ X5 )
& ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ nat @ ( F3 @ Y6 ) @ ( F3 @ X5 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_1738_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X6: set @ A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ X6 )
& ( ord_less @ A @ X5 @ Xa ) ) )
=> ~ ( finite_finite2 @ A @ X6 ) ) ) ) ).
% infinite_growing
thf(fact_1739_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ S3 )
& ~ ? [Xa: A] :
( ( member @ A @ Xa @ S3 )
& ( ord_less @ A @ Xa @ X5 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1740_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K2: A,F3: A > nat,N: nat] :
( ( P @ K2 )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ? [Y6: A] :
( ( P @ Y6 )
& ~ ( ord_less_eq @ nat @ ( F3 @ Y6 ) @ ( F3 @ X5 ) ) ) )
=> ? [Y4: A] :
( ( P @ Y4 )
& ~ ( ord_less @ nat @ ( F3 @ Y4 ) @ ( plus_plus @ nat @ ( F3 @ K2 ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_1741_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_1742_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_1743_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_1744_option_Osize__gen_I2_J,axiom,
! [A: $tType,X3: A > nat,X2: A] :
( ( size_option @ A @ X3 @ ( some @ A @ X2 ) )
= ( plus_plus @ nat @ ( X3 @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_1745_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_1746_set__decode__Suc,axiom,
! [N: nat,X3: nat] :
( ( member @ nat @ ( suc @ N ) @ ( nat_set_decode @ X3 ) )
= ( member @ nat @ N @ ( nat_set_decode @ ( divide_divide @ nat @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_1747_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R2: A,A3: A,B2: A,C3: A,D3: A] :
( ( R2
!= ( zero_zero @ A ) )
=> ( ( ( A3 = B2 )
& ( C3 != D3 ) )
=> ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R2 @ C3 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R2 @ D3 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1748_nat__dvd__1__iff__1,axiom,
! [M2: nat] :
( ( dvd_dvd @ nat @ M2 @ ( one_one @ nat ) )
= ( M2
= ( one_one @ nat ) ) ) ).
% nat_dvd_1_iff_1
thf(fact_1749_dvd__1__left,axiom,
! [K2: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K2 ) ).
% dvd_1_left
thf(fact_1750_dvd__1__iff__1,axiom,
! [M2: nat] :
( ( dvd_dvd @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( M2
= ( suc @ ( zero_zero @ nat ) ) ) ) ).
% dvd_1_iff_1
thf(fact_1751_dvd__add__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_triv_right_iff
thf(fact_1752_dvd__add__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_triv_left_iff
thf(fact_1753_set__decode__inverse,axiom,
! [N: nat] :
( ( nat_set_encode @ ( nat_set_decode @ N ) )
= N ) ).
% set_decode_inverse
thf(fact_1754_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ ( times_times @ A @ C3 @ A3 ) @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1755_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C3 @ A3 ) ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1756_div__add,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ) ).
% div_add
thf(fact_1757_signed__take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_Suc_1
thf(fact_1758_signed__take__bit__numeral__of__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K2: num] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( numeral_numeral @ nat @ K2 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_numeral_of_1
thf(fact_1759_dbl__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] :
( ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K2 ) )
= ( numeral_numeral @ A @ ( bit0 @ K2 ) ) ) ) ).
% dbl_simps(5)
thf(fact_1760_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_1761_set__encode__inverse,axiom,
! [A6: set @ nat] :
( ( finite_finite2 @ nat @ A6 )
=> ( ( nat_set_decode @ ( nat_set_encode @ A6 ) )
= A6 ) ) ).
% set_encode_inverse
thf(fact_1762_even__Suc,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% even_Suc
thf(fact_1763_even__Suc__Suc__iff,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ N ) ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% even_Suc_Suc_iff
thf(fact_1764_even__mult__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_mult_iff
thf(fact_1765_even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_add
thf(fact_1766_odd__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) ) )
= ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
!= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ) ).
% odd_add
thf(fact_1767_even__mod__2__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ).
% even_mod_2_iff
thf(fact_1768_odd__Suc__div__two,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( divide_divide @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% odd_Suc_div_two
thf(fact_1769_even__Suc__div__two,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( divide_divide @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% even_Suc_div_two
thf(fact_1770_signed__take__bit__Suc__bit0,axiom,
! [N: nat,K2: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit0 @ K2 ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_1771_set__decode__0,axiom,
! [X3: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X3 ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X3 ) ) ) ).
% set_decode_0
thf(fact_1772_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_1773_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_1774_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_1775_even__plus__one__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_plus_one_iff
thf(fact_1776_even__diff,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ).
% even_diff
thf(fact_1777_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_1778_even__diff__nat,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M2 @ N ) )
= ( ( ord_less @ nat @ M2 @ N )
| ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ).
% even_diff_nat
thf(fact_1779_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) )
= ( ( ( numeral_numeral @ nat @ W )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_1780_even__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_two
thf(fact_1781_odd__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% odd_succ_div_two
thf(fact_1782_even__succ__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_2
thf(fact_1783_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_1784_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_1785_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
= A3 ) ) ) ).
% odd_two_times_div_two_succ
thf(fact_1786_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W ) )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_1787_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_1788_dvd__add__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% dvd_add_right_iff
thf(fact_1789_dvd__add__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C3 )
=> ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% dvd_add_left_iff
thf(fact_1790_dvd__add,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ A3 @ C3 )
=> ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ) ).
% dvd_add
thf(fact_1791_dvd__power__same,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X3: A,Y: A,N: nat] :
( ( dvd_dvd @ A @ X3 @ Y )
=> ( dvd_dvd @ A @ ( power_power @ A @ X3 @ N ) @ ( power_power @ A @ Y @ N ) ) ) ) ).
% dvd_power_same
thf(fact_1792_dvd__diff__nat,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ K2 @ M2 )
=> ( ( dvd_dvd @ nat @ K2 @ N )
=> ( dvd_dvd @ nat @ K2 @ ( minus_minus @ nat @ M2 @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_1793_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ A3 ) )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ B2 ) ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% subset_divisors_dvd
thf(fact_1794_even__signed__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ M2 @ A3 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ).
% even_signed_take_bit_iff
thf(fact_1795_finite__set__decode,axiom,
! [N: nat] : ( finite_finite2 @ nat @ ( nat_set_decode @ N ) ) ).
% finite_set_decode
thf(fact_1796_div__plus__div__distrib__dvd__left,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_1797_div__plus__div__distrib__dvd__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_1798_div__power,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,N: nat] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( power_power @ A @ ( divide_divide @ A @ A3 @ B2 ) @ N )
= ( divide_divide @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% div_power
thf(fact_1799_le__imp__power__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% le_imp_power_dvd
thf(fact_1800_power__le__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat,B2: A,M2: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ B2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M2 ) @ B2 ) ) ) ) ).
% power_le_dvd
thf(fact_1801_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X3: A,Y: A,N: nat,M2: nat] :
( ( dvd_dvd @ A @ X3 @ Y )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( dvd_dvd @ A @ ( power_power @ A @ X3 @ N ) @ ( power_power @ A @ Y @ M2 ) ) ) ) ) ).
% dvd_power_le
thf(fact_1802_nat__dvd__not__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ N )
=> ~ ( dvd_dvd @ nat @ N @ M2 ) ) ) ).
% nat_dvd_not_less
thf(fact_1803_dvd__minus__self,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd @ nat @ M2 @ ( minus_minus @ nat @ N @ M2 ) )
= ( ( ord_less @ nat @ N @ M2 )
| ( dvd_dvd @ nat @ M2 @ N ) ) ) ).
% dvd_minus_self
thf(fact_1804_dvd__diffD,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ K2 @ ( minus_minus @ nat @ M2 @ N ) )
=> ( ( dvd_dvd @ nat @ K2 @ N )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( dvd_dvd @ nat @ K2 @ M2 ) ) ) ) ).
% dvd_diffD
thf(fact_1805_dvd__diffD1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ K2 @ ( minus_minus @ nat @ M2 @ N ) )
=> ( ( dvd_dvd @ nat @ K2 @ M2 )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( dvd_dvd @ nat @ K2 @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_1806_less__eq__dvd__minus,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( dvd_dvd @ nat @ M2 @ N )
= ( dvd_dvd @ nat @ M2 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1807_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X4: A] : ( plus_plus @ A @ X4 @ X4 ) ) ) ) ).
% dbl_def
thf(fact_1808_even__numeral,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: num] : ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ N ) ) ) ) ).
% even_numeral
thf(fact_1809_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_1810_dvd__imp__le,axiom,
! [K2: nat,N: nat] :
( ( dvd_dvd @ nat @ K2 @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat @ K2 @ N ) ) ) ).
% dvd_imp_le
thf(fact_1811_dvd__mult__cancel,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( dvd_dvd @ nat @ M2 @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_1812_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X4: nat] :
( collect @ nat
@ ^ [N3: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_1813_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M2: nat,Q3: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( ( modulo_modulo @ nat @ M2 @ Q3 )
= ( modulo_modulo @ nat @ N @ Q3 ) )
= ( dvd_dvd @ nat @ Q3 @ ( minus_minus @ nat @ M2 @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_1814_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_1815_evenE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B4: A] :
( A3
!= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B4 ) ) ) ) ).
% evenE
thf(fact_1816_odd__even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% odd_even_add
thf(fact_1817_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_1818_bit__eq__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [A8: A,B8: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A8 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B8 ) )
& ( ( divide_divide @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ B8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_1819_subset__decode__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ ( nat_set_decode @ M2 ) @ ( nat_set_decode @ N ) )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% subset_decode_imp_le
thf(fact_1820_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X3: A,M2: nat,N: nat] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X3 @ M2 ) @ ( power_power @ A @ X3 @ N ) )
= ( ( dvd_dvd @ A @ X3 @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_1821_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat,X3: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
| ( X3
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X3 @ ( power_power @ A @ X3 @ N ) ) ) ) ).
% dvd_power
thf(fact_1822_div2__even__ext__nat,axiom,
! [X3: nat,Y: nat] :
( ( ( divide_divide @ nat @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X3 )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Y ) )
=> ( X3 = Y ) ) ) ).
% div2_even_ext_nat
thf(fact_1823_dvd__mult__cancel1,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ M2 @ N ) @ M2 )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel1
thf(fact_1824_dvd__mult__cancel2,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ N @ M2 ) @ M2 )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel2
thf(fact_1825_dvd__minus__add,axiom,
! [Q3: nat,N: nat,R2: nat,M2: nat] :
( ( ord_less_eq @ nat @ Q3 @ N )
=> ( ( ord_less_eq @ nat @ Q3 @ ( times_times @ nat @ R2 @ M2 ) )
=> ( ( dvd_dvd @ nat @ M2 @ ( minus_minus @ nat @ N @ Q3 ) )
= ( dvd_dvd @ nat @ M2 @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R2 @ M2 ) @ Q3 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_1826_power__dvd__imp__le,axiom,
! [I: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M2 ) @ ( power_power @ nat @ I @ N ) )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_1827_mod__nat__eqI,axiom,
! [R2: nat,N: nat,M2: nat] :
( ( ord_less @ nat @ R2 @ N )
=> ( ( ord_less_eq @ nat @ R2 @ M2 )
=> ( ( dvd_dvd @ nat @ N @ ( minus_minus @ nat @ M2 @ R2 ) )
=> ( ( modulo_modulo @ nat @ M2 @ N )
= R2 ) ) ) ) ).
% mod_nat_eqI
thf(fact_1828_even__two__times__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= A3 ) ) ) ).
% even_two_times_div_two
thf(fact_1829_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_1830_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_1831_power__mono__odd,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A,B2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono_odd
thf(fact_1832_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_1833_dvd__power__iff__le,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ( ( dvd_dvd @ nat @ ( power_power @ nat @ K2 @ M2 ) @ ( power_power @ nat @ K2 @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_1834_signed__take__bit__int__less__exp,axiom,
! [N: nat,K2: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% signed_take_bit_int_less_exp
thf(fact_1835_even__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M2 @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% even_unset_bit_iff
thf(fact_1836_even__set__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M2 @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( M2
!= ( zero_zero @ nat ) ) ) ) ) ).
% even_set_bit_iff
thf(fact_1837_even__flip__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M2 @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
!= ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% even_flip_bit_iff
thf(fact_1838_even__diff__iff,axiom,
! [K2: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ int @ K2 @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K2 @ L ) ) ) ).
% even_diff_iff
thf(fact_1839_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B4: A] :
( A3
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B4 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_1840_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_1841_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_1842_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_1843_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_1844_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_1845_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq @ int @ K2 @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) )
= ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_1846_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) @ K2 )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K2 ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_1847_even__set__encode__iff,axiom,
! [A6: set @ nat] :
( ( finite_finite2 @ nat @ A6 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A6 ) )
= ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A6 ) ) ) ) ).
% even_set_encode_iff
thf(fact_1848_add__0__iff,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [B2: A,A3: A] :
( ( B2
= ( plus_plus @ A @ B2 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_1849_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ( A3 != B2 )
& ( C3 != D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A3 @ D3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_1850_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W: A,Y: A,X3: A,Z2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W @ Y ) @ ( times_times @ A @ X3 @ Z2 ) )
= ( plus_plus @ A @ ( times_times @ A @ W @ Z2 ) @ ( times_times @ A @ X3 @ Y ) ) )
= ( ( W = X3 )
| ( Y = Z2 ) ) ) ) ).
% crossproduct_eq
thf(fact_1851_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_1852_signed__take__bit__int__less__eq,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K2 )
=> ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) @ ( minus_minus @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_1853_even__mask__div__iff_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M2: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% even_mask_div_iff'
thf(fact_1854_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_1855_option_Osize__gen_I1_J,axiom,
! [A: $tType,X3: A > nat] :
( ( size_option @ A @ X3 @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size_gen(1)
thf(fact_1856_even__mask__div__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% even_mask_div_iff
thf(fact_1857_set__decode__plus__power__2,axiom,
! [N: nat,Z2: nat] :
( ~ ( member @ nat @ N @ ( nat_set_decode @ Z2 ) )
=> ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z2 ) )
= ( insert2 @ nat @ N @ ( nat_set_decode @ Z2 ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_1858_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ord_less @ nat @ N @ M2 )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M2 @ N )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_1859_pow__divides__pow__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% pow_divides_pow_iff
thf(fact_1860_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_1861_even__even__mod__4__iff,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_even_mod_4_iff
thf(fact_1862_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D4: A,T2: A] :
( ( dvd_dvd @ A @ D3 @ D4 )
=> ! [X: A,K4: A] :
( ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ K4 @ D4 ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_1863_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D4: A,T2: A] :
( ( dvd_dvd @ A @ D3 @ D4 )
=> ! [X: A,K4: A] :
( ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X @ T2 ) )
= ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ K4 @ D4 ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_1864_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X4: A] : ( P @ ( times_times @ A @ L @ X4 ) ) )
= ( ? [X4: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X4 @ ( zero_zero @ A ) ) )
& ( P @ X4 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_1865_vebt__buildup_Oelims,axiom,
! [X3: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X3 )
= Y )
=> ( ( ( X3
= ( zero_zero @ nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X3
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va3: nat] :
( ( X3
= ( suc @ ( suc @ Va3 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y
= ( 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 ) ) )
=> ( Y
= ( 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.elims
thf(fact_1866_intind,axiom,
! [A: $tType,I: nat,N: nat,P: A > $o,X3: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( P @ X3 )
=> ( P @ ( nth @ A @ ( replicate @ A @ N @ X3 ) @ I ) ) ) ) ).
% intind
thf(fact_1867_length__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( size_size @ ( list @ A ) @ ( replicate @ A @ N @ X3 ) )
= N ) ).
% length_replicate
thf(fact_1868_in__set__replicate,axiom,
! [A: $tType,X3: A,N: nat,Y: A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( replicate @ A @ N @ Y ) ) )
= ( ( X3 = Y )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_1869_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
& ( P @ X4 ) ) )
= ( ( P @ A3 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_1870_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
=> ( P @ X4 ) ) )
= ( ( P @ A3 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_1871_set__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X3 ) )
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_1872_dvd__antisym,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd @ nat @ M2 @ N )
=> ( ( dvd_dvd @ nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% dvd_antisym
thf(fact_1873_replicate__length__same,axiom,
! [A: $tType,Xs2: list @ A,X3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( X5 = X3 ) )
=> ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs2 ) @ X3 )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_1874_replicate__eqI,axiom,
! [A: $tType,Xs2: list @ A,N: nat,X3: A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= N )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ ( set2 @ A @ Xs2 ) )
=> ( Y4 = X3 ) )
=> ( Xs2
= ( replicate @ A @ N @ X3 ) ) ) ) ).
% replicate_eqI
thf(fact_1875_set__replicate__Suc,axiom,
! [A: $tType,N: nat,X3: A] :
( ( set2 @ A @ ( replicate @ A @ ( suc @ N ) @ X3 ) )
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_replicate_Suc
thf(fact_1876_set__replicate__conv__if,axiom,
! [A: $tType,N: nat,X3: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X3 ) )
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_1877_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z3 )
=> ~ ( ord_less_eq @ A @ T2 @ X ) ) ) ).
% minf(8)
thf(fact_1878_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z3 )
=> ( ord_less_eq @ A @ X @ T2 ) ) ) ).
% minf(6)
thf(fact_1879_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ Z3 @ X )
=> ( ord_less_eq @ A @ T2 @ X ) ) ) ).
% pinf(8)
thf(fact_1880_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ Z3 @ X )
=> ~ ( ord_less_eq @ A @ X @ T2 ) ) ) ).
% pinf(6)
thf(fact_1881_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B2: nat] :
( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
= ( P @ B4 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ ( zero_zero @ nat ) )
=> ( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
=> ( P @ A5 @ ( plus_plus @ nat @ A5 @ B4 ) ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1882_bezout__lemma__nat,axiom,
! [D3: nat,A3: nat,B2: nat,X3: nat,Y: nat] :
( ( dvd_dvd @ nat @ D3 @ A3 )
=> ( ( dvd_dvd @ nat @ D3 @ B2 )
=> ( ( ( ( times_times @ nat @ A3 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y ) @ D3 ) )
| ( ( times_times @ nat @ B2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y ) @ D3 ) ) )
=> ? [X5: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D3 @ A3 )
& ( dvd_dvd @ nat @ D3 @ ( plus_plus @ nat @ A3 @ B2 ) )
& ( ( ( times_times @ nat @ A3 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ Y4 ) @ D3 ) )
| ( ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y4 ) @ D3 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1883_bezout__add__nat,axiom,
! [A3: nat,B2: nat] :
? [D2: nat,X5: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D2 @ A3 )
& ( dvd_dvd @ nat @ D2 @ B2 )
& ( ( ( times_times @ nat @ A3 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D2 ) )
| ( ( times_times @ nat @ B2 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y4 ) @ D2 ) ) ) ) ).
% bezout_add_nat
thf(fact_1884_vebt__buildup_Osimps_I3_J,axiom,
! [Va: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.simps(3)
thf(fact_1885_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X: B] :
( ( ord_less @ B @ X @ Z3 )
=> ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X @ S ) ) ) ) ) ) ).
% minf(10)
thf(fact_1886_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X: B] :
( ( ord_less @ B @ X @ Z3 )
=> ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X @ S ) )
= ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X @ S ) ) ) ) ) ).
% minf(9)
thf(fact_1887_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X: B] :
( ( ord_less @ B @ Z3 @ X )
=> ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X @ S ) ) ) ) ) ) ).
% pinf(10)
thf(fact_1888_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X: B] :
( ( ord_less @ B @ Z3 @ X )
=> ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X @ S ) )
= ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X @ S ) ) ) ) ) ).
% pinf(9)
thf(fact_1889_bezout__add__strong__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [D2: nat,X5: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D2 @ A3 )
& ( dvd_dvd @ nat @ D2 @ B2 )
& ( ( times_times @ nat @ A3 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D2 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1890_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N3: nat,A8: A] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_1891_vebt__buildup_Opelims,axiom,
! [X3: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X3 )
= Y )
=> ( ( accp @ nat @ vEBT_v4011308405150292612up_rel @ X3 )
=> ( ( ( X3
= ( zero_zero @ nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X3
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va3: nat] :
( ( X3
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y
= ( 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 ) ) )
=> ( Y
= ( 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 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.pelims
thf(fact_1892_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_1893_artanh__def,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A )
= ( ^ [X4: A] : ( divide_divide @ A @ ( ln_ln @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X4 ) @ ( minus_minus @ A @ ( one_one @ A ) @ X4 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% artanh_def
thf(fact_1894_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L2: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q4: 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 ) ) @ Q4 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ R5 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_1895_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K2: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit1 @ K2 ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_1896_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N3: nat,A8: A] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_1897_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
= A3 ) ) ).
% add.inverse_inverse
thf(fact_1898_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B2 ) )
= ( A3 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_1899_Compl__subset__Compl__iff,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1900_Compl__anti__mono,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) ) ).
% Compl_anti_mono
thf(fact_1901_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% compl_le_compl_iff
thf(fact_1902_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_1903_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_1904_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_1905_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_1906_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_1907_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_1908_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% neg_less_iff_less
thf(fact_1909_neg__numeral__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M2: num,N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( M2 = N ) ) ) ).
% neg_numeral_eq_iff
thf(fact_1910_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_add_distrib
thf(fact_1911_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) )
= B2 ) ) ).
% minus_add_cancel
thf(fact_1912_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) )
= B2 ) ) ).
% add_minus_cancel
thf(fact_1913_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% minus_diff_eq
thf(fact_1914_semiring__norm_I88_J,axiom,
! [M2: num,N: num] :
( ( bit0 @ M2 )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_1915_semiring__norm_I89_J,axiom,
! [M2: num,N: num] :
( ( bit1 @ M2 )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_1916_semiring__norm_I84_J,axiom,
! [N: num] :
( one2
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_1917_semiring__norm_I86_J,axiom,
! [M2: num] :
( ( bit1 @ M2 )
!= one2 ) ).
% semiring_norm(86)
thf(fact_1918_Compl__disjoint2,axiom,
! [A: $tType,A6: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ A6 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint2
thf(fact_1919_Compl__disjoint,axiom,
! [A: $tType,A6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint
thf(fact_1920_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_1921_Diff__Compl,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ).
% Diff_Compl
thf(fact_1922_Compl__Diff__eq,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ B5 ) ) ).
% Compl_Diff_eq
thf(fact_1923_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F3: B > C > A,A3: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F3 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( F3 @ A3 @ B2 ) ) ).
% case_prod_conv
thf(fact_1924_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_1925_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_1926_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_1927_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_1928_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_1929_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_1930_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_1931_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_1932_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_1933_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_1934_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_1935_add__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_1936_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( plus_plus @ A @ A3 @ B2 ) ) ) ).
% diff_minus_eq_add
thf(fact_1937_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% uminus_add_conv_diff
thf(fact_1938_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_1939_boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ X3 @ ( uminus_uminus @ A @ X3 ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1940_boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ X3 )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1941_inf__compl__bot__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y @ ( uminus_uminus @ A @ X3 ) ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_right
thf(fact_1942_inf__compl__bot__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left2
thf(fact_1943_inf__compl__bot__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ ( inf_inf @ A @ X3 @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left1
thf(fact_1944_subset__Compl__singleton,axiom,
! [A: $tType,A6: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A6 ) ) ) ).
% subset_Compl_singleton
thf(fact_1945_take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_Suc_1
thf(fact_1946_take__bit__numeral__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_numeral_1
thf(fact_1947_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_1948_semiring__norm_I7_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% semiring_norm(7)
thf(fact_1949_semiring__norm_I9_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% semiring_norm(9)
thf(fact_1950_semiring__norm_I15_J,axiom,
! [M2: num,N: num] :
( ( times_times @ num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( bit0 @ ( times_times @ num @ ( bit1 @ M2 ) @ N ) ) ) ).
% semiring_norm(15)
thf(fact_1951_semiring__norm_I14_J,axiom,
! [M2: num,N: num] :
( ( times_times @ num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( bit0 @ ( times_times @ num @ M2 @ ( bit1 @ N ) ) ) ) ).
% semiring_norm(14)
thf(fact_1952_semiring__norm_I72_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq @ num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M2 @ N ) ) ).
% semiring_norm(72)
thf(fact_1953_semiring__norm_I81_J,axiom,
! [M2: num,N: num] :
( ( ord_less @ num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M2 @ N ) ) ).
% semiring_norm(81)
thf(fact_1954_semiring__norm_I70_J,axiom,
! [M2: num] :
~ ( ord_less_eq @ num @ ( bit1 @ M2 ) @ one2 ) ).
% semiring_norm(70)
thf(fact_1955_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_1956_dbl__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] :
( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K2 ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K2 ) ) ) ) ) ).
% dbl_simps(1)
thf(fact_1957_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_1958_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_1959_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_1960_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_1961_minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) )
= ( one_one @ A ) ) ) ).
% minus_one_mult_self
thf(fact_1962_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ A3 ) )
= A3 ) ) ).
% left_minus_one_mult_self
thf(fact_1963_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_1964_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_1965_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_1966_semiring__norm_I168_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W ) ) ) @ Y ) ) ) ).
% semiring_norm(168)
thf(fact_1967_diff__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_1968_diff__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ).
% diff_numeral_simps(2)
thf(fact_1969_zdiv__numeral__Bit1,axiom,
! [V2: num,W: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit1 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W ) ) ) ).
% zdiv_numeral_Bit1
thf(fact_1970_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_1971_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% semiring_norm(4)
thf(fact_1972_semiring__norm_I5_J,axiom,
! [M2: num] :
( ( plus_plus @ num @ ( bit0 @ M2 ) @ one2 )
= ( bit1 @ M2 ) ) ).
% semiring_norm(5)
thf(fact_1973_semiring__norm_I8_J,axiom,
! [M2: num] :
( ( plus_plus @ num @ ( bit1 @ M2 ) @ one2 )
= ( bit0 @ ( plus_plus @ num @ M2 @ one2 ) ) ) ).
% semiring_norm(8)
thf(fact_1974_semiring__norm_I10_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ ( plus_plus @ num @ M2 @ N ) @ one2 ) ) ) ).
% semiring_norm(10)
thf(fact_1975_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W: num,Y: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Y ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) ) @ Y ) ) ) ).
% semiring_norm(170)
thf(fact_1976_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W: num,Y: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) ) @ Y ) ) ) ).
% semiring_norm(171)
thf(fact_1977_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W: num,Y: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) @ Y ) ) ) ).
% semiring_norm(172)
thf(fact_1978_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M2 @ N ) ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_1979_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M2 @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_1980_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M2 @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_1981_neg__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N: num] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less_eq @ num @ N @ M2 ) ) ) ).
% neg_numeral_le_iff
thf(fact_1982_neg__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less @ num @ N @ M2 ) ) ) ).
% neg_numeral_less_iff
thf(fact_1983_semiring__norm_I16_J,axiom,
! [M2: num,N: num] :
( ( times_times @ num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus @ num @ ( plus_plus @ num @ M2 @ N ) @ ( bit0 @ ( times_times @ num @ M2 @ N ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_1984_semiring__norm_I74_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq @ num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M2 @ N ) ) ).
% semiring_norm(74)
thf(fact_1985_semiring__norm_I79_J,axiom,
! [M2: num,N: num] :
( ( ord_less @ num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M2 @ N ) ) ).
% semiring_norm(79)
thf(fact_1986_not__neg__one__le__neg__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) )
= ( M2 != one2 ) ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_1987_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B2 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_1988_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_1989_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,W: num] :
( ( A3
= ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= B2 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1990_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= A3 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1991_neg__numeral__less__neg__one__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( M2 != one2 ) ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_1992_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B2 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_1993_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_1994_power2__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_minus
thf(fact_1995_odd__of__bool__self,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [P2: $o] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_neq_one_of_bool @ A @ P2 ) ) )
= P2 ) ) ).
% odd_of_bool_self
thf(fact_1996_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_1997_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_1998_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_1999_minus__1__div__2__eq,axiom,
! [A: $tType] :
( ( euclid8789492081693882211th_nat @ A )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% minus_1_div_2_eq
thf(fact_2000_minus__1__mod__2__eq,axiom,
! [A: $tType] :
( ( euclid8789492081693882211th_nat @ A )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% minus_1_mod_2_eq
thf(fact_2001_bits__minus__1__mod__2__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% bits_minus_1_mod_2_eq
thf(fact_2002_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_2003_of__bool__half__eq__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [B2: $o] :
( ( divide_divide @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ).
% of_bool_half_eq_0
thf(fact_2004_Parity_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_2005_power__minus__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_minus_odd
thf(fact_2006_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_2007_Suc__div__eq__add3__div__numeral,axiom,
! [M2: nat,V2: num] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral @ nat @ V2 ) )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ ( numeral_numeral @ nat @ V2 ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_2008_div__Suc__eq__div__add3,axiom,
! [M2: nat,N: nat] :
( ( divide_divide @ nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( divide_divide @ nat @ M2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_2009_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_2010_diff__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ one2 ) ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_2011_Suc__mod__eq__add3__mod__numeral,axiom,
! [M2: nat,V2: num] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral @ nat @ V2 ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ ( numeral_numeral @ nat @ V2 ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_2012_mod__Suc__eq__mod__add3,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo @ nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( modulo_modulo @ nat @ M2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% mod_Suc_eq_mod_add3
thf(fact_2013_signed__take__bit__Suc__minus__bit0,axiom,
! [N: nat,K2: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K2 ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_2014_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_2015_power__minus1__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( one_one @ A ) ) ) ).
% power_minus1_even
thf(fact_2016_neg__one__even__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( one_one @ A ) ) ) ) ).
% neg_one_even_power
thf(fact_2017_neg__one__odd__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% neg_one_odd_power
thf(fact_2018_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_2019_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_2020_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_2021_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_2022_take__bit__of__exp,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_of_exp
thf(fact_2023_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_2024_zmod__numeral__Bit1,axiom,
! [V2: num,W: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W ) ) ) @ ( one_one @ int ) ) ) ).
% zmod_numeral_Bit1
thf(fact_2025_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_2026_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K2: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K2 ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_2027_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% equation_minus_iff
thf(fact_2028_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A3 ) ) ) ).
% minus_equation_iff
thf(fact_2029_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( H @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X15: A,X23: B] : ( H @ ( F3 @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_2030_power__minus__Bit1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: A,K2: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit1 @ K2 ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit1 @ K2 ) ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_2031_take__bit__add,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ).
% take_bit_add
thf(fact_2032_take__bit__tightened,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A,B2: A,M2: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M2 @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ M2 @ B2 ) ) ) ) ) ).
% take_bit_tightened
thf(fact_2033_take__bit__tightened__less__eq__nat,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M2 @ Q3 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N @ Q3 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_2034_take__bit__nat__less__eq__self,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M2 ) @ M2 ) ).
% take_bit_nat_less_eq_self
thf(fact_2035_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_2036_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X3 ) )
=> ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_2037_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% compl_mono
thf(fact_2038_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_imp_neg_le
thf(fact_2039_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% minus_le_iff
thf(fact_2040_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_minus_iff
thf(fact_2041_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% less_minus_iff
thf(fact_2042_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% minus_less_iff
thf(fact_2043_verit__eq__simplify_I14_J,axiom,
! [X2: num,X32: num] :
( ( bit0 @ X2 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_2044_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one2
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_2045_numeral__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M2: num,N: num] :
( ( numeral_numeral @ A @ M2 )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_2046_neg__numeral__neq__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M2: num,N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) )
!= ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_neq_numeral
thf(fact_2047_is__num__normalize_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% is_num_normalize(8)
thf(fact_2048_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_2049_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A6: A,K2: A,A3: A] :
( ( A6
= ( plus_plus @ A @ K2 @ A3 ) )
=> ( ( uminus_uminus @ A @ A6 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_2050_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A3 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% minus_diff_commute
thf(fact_2051_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > B > C,X1: A,X2: B] :
( ( product_case_prod @ A @ B @ C @ F3 @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= ( F3 @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_2052_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z2: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z2 ) )
=> ~ ! [X5: B,Y4: C] :
( ( Z2
= ( product_Pair @ B @ C @ X5 @ Y4 ) )
=> ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% case_prodE2
thf(fact_2053_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X4: A,Y3: B] : ( F3 @ ( product_Pair @ A @ B @ X4 @ Y3 ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_2054_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B > C,G3: ( product_prod @ A @ B ) > C] :
( ! [X5: A,Y4: B] :
( ( F3 @ X5 @ Y4 )
= ( G3 @ ( product_Pair @ A @ B @ X5 @ Y4 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F3 )
= G3 ) ) ).
% cond_case_prod_eta
thf(fact_2055_Collect__imp__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) @ ( collect @ A @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_2056_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_2057_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_2058_take__bit__tightened__less__eq__int,axiom,
! [M2: nat,N: nat,K2: int] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M2 @ K2 ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_2059_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( ( bit_ri4674362597316999326ke_bit @ A @ N @ A3 )
= ( bit_ri4674362597316999326ke_bit @ A @ N @ B2 ) )
= ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ B2 ) ) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
thf(fact_2060_signed__take__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N @ M2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N ) @ ( bit_ri4674362597316999326ke_bit @ A @ M2 ) @ A3 ) ) ) ).
% signed_take_bit_take_bit
thf(fact_2061_not__numeral__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_2062_neg__numeral__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_le_numeral
thf(fact_2063_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_2064_neg__numeral__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_less_numeral
thf(fact_2065_not__numeral__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_2066_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one2 )
=> ( ! [X22: num] :
( Y
!= ( bit0 @ X22 ) )
=> ~ ! [X33: num] :
( Y
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_2067_xor__num_Ocases,axiom,
! [X3: product_prod @ num @ num] :
( ( X3
!= ( product_Pair @ num @ num @ one2 @ one2 ) )
=> ( ! [N2: num] :
( X3
!= ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) )
=> ( ! [N2: num] :
( X3
!= ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) )
=> ( ! [M: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit0 @ M ) @ one2 ) )
=> ( ! [M: num,N2: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit0 @ M ) @ ( bit0 @ N2 ) ) )
=> ( ! [M: num,N2: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit0 @ M ) @ ( bit1 @ N2 ) ) )
=> ( ! [M: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit1 @ M ) @ one2 ) )
=> ( ! [M: num,N2: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) ) )
=> ~ ! [M: num,N2: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit1 @ M ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_2068_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_2069_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_2070_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add_eq_0_iff
thf(fact_2071_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_2072_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A3 )
= B2 ) ) ) ).
% add.inverse_unique
thf(fact_2073_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( uminus_uminus @ A @ B2 ) )
= ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_2074_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= B2 )
= ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_2075_numeral__times__minus__swap,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [W: num,X3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ ( uminus_uminus @ A @ X3 ) )
= ( times_times @ A @ X3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_2076_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_2077_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_2078_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A8: A,B8: A] : ( plus_plus @ A @ A8 @ ( uminus_uminus @ A @ B8 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_2079_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A8: A,B8: A] : ( plus_plus @ A @ A8 @ ( uminus_uminus @ A @ B8 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_2080_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B5: A,K2: A,B2: A,A3: A] :
( ( B5
= ( plus_plus @ A @ K2 @ B2 ) )
=> ( ( minus_minus @ A @ A3 @ B5 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K2 ) @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_2081_take__bit__unset__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M2: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M2 @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M2 @ A3 ) )
= ( bit_se2638667681897837118et_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_2082_take__bit__set__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M2: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M2 @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M2 @ A3 ) )
= ( bit_se5668285175392031749et_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_2083_take__bit__flip__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M2: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M2 @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M2 @ A3 ) )
= ( bit_se8732182000553998342ip_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_2084_inf__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ A3 ) @ ( inf_inf @ A @ X3 @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left2
thf(fact_2085_inf__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ A3 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left1
thf(fact_2086_subset__Compl__self__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_2087_Compl__Int,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) ) ) ).
% Compl_Int
thf(fact_2088_Compl__Un,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) ) ) ).
% Compl_Un
thf(fact_2089_Diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ A7 @ ( uminus_uminus @ ( set @ A ) @ B6 ) ) ) ) ).
% Diff_eq
thf(fact_2090_take__bit__Suc__minus__bit0,axiom,
! [N: nat,K2: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K2 ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_2091_take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_ri4674362597316999326ke_bit @ A @ N @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ M2 @ A3 ) ) ) ) ).
% take_bit_signed_take_bit
thf(fact_2092_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_2093_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_2094_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_2095_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_2096_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_2097_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_2098_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_2099_not__one__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_one_le_neg_numeral
thf(fact_2100_not__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_le_neg_one
thf(fact_2101_neg__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% neg_numeral_le_neg_one
thf(fact_2102_neg__one__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M2 ) ) ) ).
% neg_one_le_numeral
thf(fact_2103_neg__numeral__le__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_le_one
thf(fact_2104_neg__numeral__less__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_less_one
thf(fact_2105_neg__one__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M2 ) ) ) ).
% neg_one_less_numeral
thf(fact_2106_not__numeral__less__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_less_neg_one
thf(fact_2107_not__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_one_less_neg_numeral
thf(fact_2108_not__neg__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_2109_mult__1s__ring__1_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) ) ).
% mult_1s_ring_1(1)
thf(fact_2110_mult__1s__ring__1_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B2: A] :
( ( times_times @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) )
= ( uminus_uminus @ A @ B2 ) ) ) ).
% mult_1s_ring_1(2)
thf(fact_2111_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_2112_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_2113_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_minus
thf(fact_2114_cong__exp__iff__simps_I10_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,Q3: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_2115_cong__exp__iff__simps_I12_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,Q3: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_2116_cong__exp__iff__simps_I13_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,Q3: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ Q3 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_2117_inf__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ( inf_inf @ A @ X3 @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% inf_shunt
thf(fact_2118_power__minus__Bit0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: A,K2: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ K2 ) ) )
= ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ K2 ) ) ) ) ) ).
% power_minus_Bit0
thf(fact_2119_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P2: A,Q3: A,R2: A] :
( ( ord_less_eq @ A @ P2 @ ( sup_sup @ A @ Q3 @ R2 ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P2 @ ( uminus_uminus @ A @ Q3 ) ) @ R2 ) ) ) ).
% sup_neg_inf
thf(fact_2120_shunt2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) @ Z2 )
= ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% shunt2
thf(fact_2121_shunt1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ Z2 )
= ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y ) @ Z2 ) ) ) ) ).
% shunt1
thf(fact_2122_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K2 ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_bit1
thf(fact_2123_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A6 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ B5 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_2124_Compl__insert,axiom,
! [A: $tType,X3: A,A6: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X3 @ A6 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_2125_take__bit__Suc__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_2126_take__bit__numeral__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ K2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ K2 ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_minus_1_eq
thf(fact_2127_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_2128_power__numeral__odd,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z2: A,W: num] :
( ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ ( bit1 @ W ) ) )
= ( times_times @ A @ ( times_times @ A @ Z2 @ ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ W ) ) ) @ ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_numeral_odd
thf(fact_2129_take__bit__minus__small__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less_eq @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K2 ) )
= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K2 ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_2130_numeral__Bit1__div__2,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% numeral_Bit1_div_2
thf(fact_2131_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B2: A,C3: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 )
= B2 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_2132_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,W: num] :
( ( ( divide_divide @ A @ B2 @ C3 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_2133_odd__numeral,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: num] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ N ) ) ) ) ).
% odd_numeral
thf(fact_2134_cong__exp__iff__simps_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q3: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(3)
thf(fact_2135_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= B2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_2136_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X3 @ Z2 ) ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X3 ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_2137_power3__eq__cube,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( times_times @ A @ ( times_times @ A @ A3 @ A3 ) @ A3 ) ) ) ).
% power3_eq_cube
thf(fact_2138_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_2139_even__minus,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( uminus_uminus @ A @ A3 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ).
% even_minus
thf(fact_2140_power2__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [X3: A,Y: A] :
( ( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( X3 = Y )
| ( X3
= ( uminus_uminus @ A @ Y ) ) ) ) ) ).
% power2_eq_iff
thf(fact_2141_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_2142_take__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K2 ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_2143_take__bit__eq__mod,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N3: nat,A8: A] : ( modulo_modulo @ A @ A8 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% take_bit_eq_mod
thf(fact_2144_take__bit__nat__eq__self__iff,axiom,
! [N: nat,M2: nat] :
( ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M2 )
= M2 )
= ( ord_less @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% take_bit_nat_eq_self_iff
thf(fact_2145_take__bit__nat__less__exp,axiom,
! [N: nat,M2: nat] : ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% take_bit_nat_less_exp
thf(fact_2146_take__bit__nat__eq__self,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M2 )
= M2 ) ) ).
% take_bit_nat_eq_self
thf(fact_2147_num_Osize_I6_J,axiom,
! [X32: num] :
( ( size_size @ num @ ( bit1 @ X32 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(6)
thf(fact_2148_take__bit__nat__def,axiom,
( ( bit_se2584673776208193580ke_bit @ nat )
= ( ^ [N3: nat,M5: nat] : ( modulo_modulo @ nat @ M5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% take_bit_nat_def
thf(fact_2149_of__bool__odd__eq__mod__2,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_bool_odd_eq_mod_2
thf(fact_2150_take__bit__int__less__exp,axiom,
! [N: nat,K2: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% take_bit_int_less_exp
thf(fact_2151_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_2152_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_2153_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_2154_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_2155_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( 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 @ B2 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_2156_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_2157_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ 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 @ W ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_2158_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_2159_take__bit__int__def,axiom,
( ( bit_se2584673776208193580ke_bit @ int )
= ( ^ [N3: nat,K3: int] : ( modulo_modulo @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% take_bit_int_def
thf(fact_2160_cong__exp__iff__simps_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q3: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q3 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(7)
thf(fact_2161_cong__exp__iff__simps_I11_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,Q3: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ Q3 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(11)
thf(fact_2162_power2__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A] :
( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( A3
= ( one_one @ A ) )
| ( A3
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% power2_eq_1_iff
thf(fact_2163_Suc__div__eq__add3__div,axiom,
! [M2: nat,N: nat] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ N ) ) ).
% Suc_div_eq_add3_div
thf(fact_2164_uminus__power__if,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_power @ A @ A3 @ N ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% uminus_power_if
thf(fact_2165_Suc__mod__eq__add3__mod,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ N ) ) ).
% Suc_mod_eq_add3_mod
thf(fact_2166_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N @ K2 ) )
= ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_2167_realpow__square__minus__le,axiom,
! [U: real,X3: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( power_power @ real @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% realpow_square_minus_le
thf(fact_2168_ln__one__minus__pos__lower__bound,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( uminus_uminus @ real @ X3 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X3 ) ) ) ) ) ).
% ln_one_minus_pos_lower_bound
thf(fact_2169_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_2170_take__bit__nat__less__self__iff,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M2 ) @ M2 )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M2 ) ) ).
% take_bit_nat_less_self_iff
thf(fact_2171_zminus1__lemma,axiom,
! [A3: int,B2: int,Q3: int,R2: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( B2
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A3 ) @ B2
@ ( product_Pair @ int @ int
@ ( if @ int
@ ( R2
= ( zero_zero @ int ) )
@ ( uminus_uminus @ int @ Q3 )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q3 ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B2 @ R2 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_2172_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A3: A] :
( ! [A5: A] :
( ( ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A5 )
=> ( P @ A5 ) )
=> ( ! [A5: A,B4: $o] :
( ( P @ A5 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B4 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A5 )
=> ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B4 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% bits_induct
thf(fact_2173_take__bit__int__greater__eq__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq @ int @ K2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) )
= ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_2174_take__bit__int__less__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) @ K2 )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K2 ) ) ).
% take_bit_int_less_self_iff
thf(fact_2175_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ 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 @ W ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_2176_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_2177_square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% square_le_1
thf(fact_2178_minus__power__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
= ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% minus_power_mult_self
thf(fact_2179_minus__one__power__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( one_one @ A ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% minus_one_power_iff
thf(fact_2180_minus__1__div__exp__eq__int,axiom,
! [N: nat] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% minus_1_div_exp_eq_int
thf(fact_2181_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K2: int] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_2182_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) @ K2 )
= ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K2 ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_2183_signed__take__bit__int__greater__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less @ int @ K2 @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) )
= ( ord_less @ int @ K2 @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_2184_exp__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M2: nat,N: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M2 @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ).
% exp_mod_exp
thf(fact_2185_take__bit__int__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K2 )
= K2 )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
& ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_2186_take__bit__int__eq__self,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ K2 )
= K2 ) ) ) ).
% take_bit_int_eq_self
thf(fact_2187_take__bit__incr__eq,axiom,
! [N: nat,K2: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K2 )
!= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K2 @ ( one_one @ int ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) ) ) ).
% take_bit_incr_eq
thf(fact_2188_power__minus1__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% power_minus1_odd
thf(fact_2189_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_2190_int__bit__induct,axiom,
! [P: int > $o,K2: int] :
( ( P @ ( zero_zero @ int ) )
=> ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( ! [K: int] :
( ( P @ K )
=> ( ( K
!= ( zero_zero @ int ) )
=> ( P @ ( times_times @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ! [K: int] :
( ( P @ K )
=> ( ( K
!= ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
=> ( P @ K2 ) ) ) ) ) ).
% int_bit_induct
thf(fact_2191_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 )
= K2 )
= ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K2 )
& ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_2192_signed__take__bit__int__eq__self,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K2 )
=> ( ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 )
= K2 ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_2193_take__bit__int__less__eq,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) @ ( minus_minus @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_2194_take__bit__int__greater__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) ) ).
% take_bit_int_greater_eq
thf(fact_2195_signed__take__bit__eq__take__bit__shift,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ ( plus_plus @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_2196_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_2197_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L2: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q4: 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 ) ) @ Q4 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_2198_ln__one__plus__pos__lower__bound,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ X3 @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) ) ) ) ) ).
% ln_one_plus_pos_lower_bound
thf(fact_2199_exp__div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 )
!= ( zero_zero @ A ) )
& ( ord_less_eq @ nat @ N @ M2 ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M2 @ N ) ) ) ) ) ).
% exp_div_exp_eq
thf(fact_2200_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L2: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q4: 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 ) ) @ Q4 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_2201_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_2202_signed__take__bit__int__greater__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less @ int @ K2 @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_2203_mod__exhaust__less__4,axiom,
! [M2: nat] :
( ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ nat ) )
| ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( one_one @ nat ) )
| ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
| ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ).
% mod_exhaust_less_4
thf(fact_2204_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_2205_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q4: A,R5: A] : ( product_Pair @ A @ A @ Q4 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_2206_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_2207_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( ( ord_less_eq @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) ) ) )
& ( ~ ( ord_less_eq @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_2208_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( ( ord_less @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) ) ) )
& ( ~ ( ord_less @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_2209_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K2: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K2 ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_2210_ComplI,axiom,
! [A: $tType,C3: A,A6: set @ A] :
( ~ ( member @ A @ C3 @ A6 )
=> ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) ) ).
% ComplI
thf(fact_2211_Compl__iff,axiom,
! [A: $tType,C3: A,A6: set @ A] :
( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= ( ~ ( member @ A @ C3 @ A6 ) ) ) ).
% Compl_iff
thf(fact_2212_Compl__eq__Compl__iff,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A6 )
= ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( A6 = B5 ) ) ).
% Compl_eq_Compl_iff
thf(fact_2213_case__prodI2,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,C3: A > B > $o] :
( ! [A5: A,B4: B] :
( ( P2
= ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( C3 @ A5 @ B4 ) )
=> ( product_case_prod @ A @ B @ $o @ C3 @ P2 ) ) ).
% case_prodI2
thf(fact_2214_case__prodI,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A3: A,B2: B] :
( ( F3 @ A3 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% case_prodI
thf(fact_2215_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z2: C,C3: A > B > ( set @ C )] :
( ! [A5: A,B4: B] :
( ( P2
= ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( member @ C @ Z2 @ ( C3 @ A5 @ B4 ) ) )
=> ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_2216_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C3: B > C > ( set @ A ),A3: B,B2: C] :
( ( member @ A @ Z2 @ ( C3 @ A3 @ B2 ) )
=> ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A3 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_2217_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C3: A > B > C > $o,X3: C] :
( ! [A5: A,B4: B] :
( ( ( product_Pair @ A @ B @ A5 @ B4 )
= P2 )
=> ( C3 @ A5 @ B4 @ X3 ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ X3 ) ) ).
% case_prodI2'
thf(fact_2218_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_2219_eq__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ( numeral_numeral @ nat @ K2 )
= ( suc @ N ) )
= ( ( pred_numeral @ K2 )
= N ) ) ).
% eq_numeral_Suc
thf(fact_2220_Suc__eq__numeral,axiom,
! [N: nat,K2: num] :
( ( ( suc @ N )
= ( numeral_numeral @ nat @ K2 ) )
= ( N
= ( pred_numeral @ K2 ) ) ) ).
% Suc_eq_numeral
thf(fact_2221_less__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( suc @ N ) )
= ( ord_less @ nat @ ( pred_numeral @ K2 ) @ N ) ) ).
% less_numeral_Suc
thf(fact_2222_less__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_less @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K2 ) )
= ( ord_less @ nat @ N @ ( pred_numeral @ K2 ) ) ) ).
% less_Suc_numeral
thf(fact_2223_pred__numeral__simps_I3_J,axiom,
! [K2: num] :
( ( pred_numeral @ ( bit1 @ K2 ) )
= ( numeral_numeral @ nat @ ( bit0 @ K2 ) ) ) ).
% pred_numeral_simps(3)
thf(fact_2224_le__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K2 ) )
= ( ord_less_eq @ nat @ N @ ( pred_numeral @ K2 ) ) ) ).
% le_Suc_numeral
thf(fact_2225_le__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( suc @ N ) )
= ( ord_less_eq @ nat @ ( pred_numeral @ K2 ) @ N ) ) ).
% le_numeral_Suc
thf(fact_2226_diff__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( suc @ N ) )
= ( minus_minus @ nat @ ( pred_numeral @ K2 ) @ N ) ) ).
% diff_numeral_Suc
thf(fact_2227_diff__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K2 ) )
= ( minus_minus @ nat @ N @ ( pred_numeral @ K2 ) ) ) ).
% diff_Suc_numeral
thf(fact_2228_max__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_max @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( suc @ N ) )
= ( suc @ ( ord_max @ nat @ ( pred_numeral @ K2 ) @ N ) ) ) ).
% max_numeral_Suc
thf(fact_2229_max__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_max @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K2 ) )
= ( suc @ ( ord_max @ nat @ N @ ( pred_numeral @ K2 ) ) ) ) ).
% max_Suc_numeral
thf(fact_2230_dvd__numeral__simp,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( unique5940410009612947441es_aux @ A @ ( unique8689654367752047608divmod @ A @ N @ M2 ) ) ) ) ).
% dvd_numeral_simp
thf(fact_2231_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num] :
( ( unique8689654367752047608divmod @ A @ M2 @ one2 )
= ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M2 ) @ ( zero_zero @ A ) ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_2232_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_2233_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_2234_signed__take__bit__numeral__bit0,axiom,
! [L: num,K2: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit0 @ K2 ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_bit0
thf(fact_2235_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q4: A,R5: A] : ( product_Pair @ A @ A @ Q4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
@ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_2236_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K2: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K2 ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_2237_signed__take__bit__numeral__bit1,axiom,
! [L: num,K2: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K2 ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_2238_ComplD,axiom,
! [A: $tType,C3: A,A6: set @ A] :
( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
=> ~ ( member @ A @ C3 @ A6 ) ) ).
% ComplD
thf(fact_2239_double__complement,axiom,
! [A: $tType,A6: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= A6 ) ).
% double_complement
thf(fact_2240_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 ) ) ) ) ) ).
% uminus_set_def
thf(fact_2241_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
~ ( P @ X4 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_2242_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ A7 ) ) ) ) ).
% Compl_eq
thf(fact_2243_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C3: B > C > ( set @ A ),P2: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P2 ) )
=> ~ ! [X5: B,Y4: C] :
( ( P2
= ( product_Pair @ B @ C @ X5 @ Y4 ) )
=> ~ ( member @ A @ Z2 @ ( C3 @ X5 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_2244_case__prodE,axiom,
! [A: $tType,B: $tType,C3: A > B > $o,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C3 @ P2 )
=> ~ ! [X5: A,Y4: B] :
( ( P2
= ( product_Pair @ A @ B @ X5 @ Y4 ) )
=> ~ ( C3 @ X5 @ Y4 ) ) ) ).
% case_prodE
thf(fact_2245_case__prodD,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A3: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( F3 @ A3 @ B2 ) ) ).
% case_prodD
thf(fact_2246_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P2: product_prod @ A @ B,Z2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ Z2 )
=> ~ ! [X5: A,Y4: B] :
( ( P2
= ( product_Pair @ A @ B @ X5 @ Y4 ) )
=> ~ ( C3 @ X5 @ Y4 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_2247_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A3: A,B2: B,C3: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A3 @ B2 ) @ C3 )
=> ( R @ A3 @ B2 @ C3 ) ) ).
% case_prodD'
thf(fact_2248_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A6: A > B > $o,B5: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A6 @ B5 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_2249_numeral__eq__Suc,axiom,
( ( numeral_numeral @ nat )
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_2250_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K3 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_2251_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M5: num,N3: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N3 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N3 ) ) ) ) ) ).
% divmod_int_def
thf(fact_2252_take__bit__numeral__minus__bit0,axiom,
! [L: num,K2: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K2 ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_2253_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M5: num,N3: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M5 ) @ ( numeral_numeral @ A @ N3 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M5 ) @ ( numeral_numeral @ A @ N3 ) ) ) ) ) ) ).
% divmod_def
thf(fact_2254_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M5: num,N3: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M5 ) @ ( numeral_numeral @ nat @ N3 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M5 ) @ ( numeral_numeral @ nat @ N3 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_2255_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X4: A] : ( minus_minus @ A @ ( plus_plus @ A @ X4 @ X4 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_2256_take__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K2 ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_2257_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M5: num,N3: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M5 @ N3 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M5 ) ) @ ( unique1321980374590559556d_step @ A @ N3 @ ( unique8689654367752047608divmod @ A @ M5 @ ( bit0 @ N3 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_2258_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K2 ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_bit1
thf(fact_2259_one__div__minus__numeral,axiom,
! [N: num] :
( ( divide_divide @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% one_div_minus_numeral
thf(fact_2260_minus__one__div__numeral,axiom,
! [N: num] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% minus_one_div_numeral
thf(fact_2261_take__bit__numeral__minus__bit1,axiom,
! [L: num,K2: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K2 ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K2 ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_2262_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) ) @ X3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_2263_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M5: nat,N3: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N3
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M5 @ N3 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M5 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q4: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q4 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M5 @ N3 ) @ N3 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_2264_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K2: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K2 ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K2 ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_2265_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_2266_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A8: A,B8: B] :
( P
& ( Q @ A8 @ B8 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_2267_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_2268_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_2269_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_2270_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_2271_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_add_abs
thf(fact_2272_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_2273_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_2274_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_2275_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_2276_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_2277_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_2278_abs__power__minus,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( abs_abs @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
= ( abs_abs @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% abs_power_minus
thf(fact_2279_pred__numeral__inc,axiom,
! [K2: num] :
( ( pred_numeral @ ( inc @ K2 ) )
= ( numeral_numeral @ nat @ K2 ) ) ).
% pred_numeral_inc
thf(fact_2280_divide__le__0__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_0_abs_iff
thf(fact_2281_zero__le__divide__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B2 ) ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_2282_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_2283_Divides_Oadjust__div__eq,axiom,
! [Q3: int,R2: int] :
( ( adjust_div @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
= ( plus_plus @ int @ Q3
@ ( zero_neq_one_of_bool @ int
@ ( R2
!= ( zero_zero @ int ) ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_2284_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_2285_power2__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( power_power @ A @ ( abs_abs @ A @ A3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_abs
thf(fact_2286_abs__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% abs_power2
thf(fact_2287_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_2288_add__neg__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M2 ) ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_2289_diff__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( numeral_numeral @ A @ ( inc @ M2 ) ) ) ) ).
% diff_numeral_special(6)
thf(fact_2290_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_2291_power__even__abs__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: num,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
=> ( ( power_power @ A @ ( abs_abs @ A @ A3 ) @ ( numeral_numeral @ nat @ W ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_even_abs_numeral
thf(fact_2292_numeral__div__minus__numeral,axiom,
! [M2: num,N: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M2 @ N ) ) ) ) ).
% numeral_div_minus_numeral
thf(fact_2293_minus__numeral__div__numeral,axiom,
! [M2: num,N: num] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M2 @ N ) ) ) ) ).
% minus_numeral_div_numeral
thf(fact_2294_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu3: A,Uv3: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_2295_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_2296_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% abs_le_D1
thf(fact_2297_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% abs_minus_commute
thf(fact_2298_power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( abs_abs @ A @ ( power_power @ A @ A3 @ N ) )
= ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) ) ) ).
% power_abs
thf(fact_2299_num__induct,axiom,
! [P: num > $o,X3: num] :
( ( P @ one2 )
=> ( ! [X5: num] :
( ( P @ X5 )
=> ( P @ ( inc @ X5 ) ) )
=> ( P @ X3 ) ) ) ).
% num_induct
thf(fact_2300_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_2301_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_2302_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_2303_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_2304_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_2305_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_2306_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_2307_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_2308_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_le_iff
thf(fact_2309_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% abs_le_D2
thf(fact_2310_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_leI
thf(fact_2311_inc_Osimps_I1_J,axiom,
( ( inc @ one2 )
= ( bit0 @ one2 ) ) ).
% inc.simps(1)
thf(fact_2312_inc_Osimps_I2_J,axiom,
! [X3: num] :
( ( inc @ ( bit0 @ X3 ) )
= ( bit1 @ X3 ) ) ).
% inc.simps(2)
thf(fact_2313_inc_Osimps_I3_J,axiom,
! [X3: num] :
( ( inc @ ( bit1 @ X3 ) )
= ( bit0 @ ( inc @ X3 ) ) ) ).
% inc.simps(3)
thf(fact_2314_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X3: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ E2 ) )
=> ( X3
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_2315_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y ) @ X3 )
= ( abs_abs @ A @ ( times_times @ A @ Y @ X3 ) ) ) ) ) ).
% abs_mult_pos
thf(fact_2316_abs__eq__mult,axiom,
! [A: $tType] :
( ( ordered_ring_abs @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
| ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
& ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
| ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ( abs_abs @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% abs_eq_mult
thf(fact_2317_abs__eq__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( abs_abs @ A @ A3 )
= B2 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
& ( ( A3 = B2 )
| ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_2318_eq__abs__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( abs_abs @ A @ B2 ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ( B2 = A3 )
| ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_2319_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_2320_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_2321_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_2322_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,A3: A,R2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ A3 ) ) @ R2 )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ R2 ) @ X3 )
& ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ A3 @ R2 ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_2323_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ C3 @ D3 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ C3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_2324_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_2325_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,A3: A,R2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ A3 ) ) @ R2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ R2 ) @ X3 )
& ( ord_less @ A @ X3 @ ( plus_plus @ A @ A3 @ R2 ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_2326_add__One,axiom,
! [X3: num] :
( ( plus_plus @ num @ X3 @ one2 )
= ( inc @ X3 ) ) ).
% add_One
thf(fact_2327_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X3 ) ) ) ) ).
% abs_add_one_gt_zero
thf(fact_2328_numeral__inc,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X3: num] :
( ( numeral_numeral @ A @ ( inc @ X3 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X3 ) @ ( one_one @ A ) ) ) ) ).
% numeral_inc
thf(fact_2329_abs__le__square__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ ( abs_abs @ A @ Y ) )
= ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_2330_abs__square__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( abs_abs @ A @ X3 )
= ( one_one @ A ) ) ) ) ).
% abs_square_eq_1
thf(fact_2331_power__even__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_even_abs
thf(fact_2332_power2__le__iff__abs__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ Y ) ) ) ) ).
% power2_le_iff_abs_le
thf(fact_2333_abs__sqrt__wlog,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A > A > $o,X3: A] :
( ! [X5: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 )
=> ( P @ X5 @ ( power_power @ A @ X5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ ( abs_abs @ A @ X3 ) @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_2334_abs__square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ ( one_one @ A ) ) ) ) ).
% abs_square_le_1
thf(fact_2335_abs__square__less__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less @ A @ ( abs_abs @ A @ X3 ) @ ( one_one @ A ) ) ) ) ).
% abs_square_less_1
thf(fact_2336_power__mono__even,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono_even
thf(fact_2337_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M5: nat,N3: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M5 @ N3 ) @ ( modulo_modulo @ nat @ M5 @ N3 ) ) ) ) ).
% divmod_nat_def
thf(fact_2338_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) ) @ X3 ) ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_2339_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( 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 ) @ X3 ) ) @ X3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound
thf(fact_2340_tanh__ln__real,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( tanh @ real @ ( ln_ln @ real @ X3 ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% tanh_ln_real
thf(fact_2341_and__int_Oelims,axiom,
! [X3: int,Xa2: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X3 @ Xa2 )
= Y )
=> ( ( ( ( member @ int @ X3 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa2 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member @ int @ X3 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa2 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_2342_and__int_Osimps,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( ( member @ int @ K3 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L2 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
@ ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) )
@ ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_2343_arctan__double,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ X3 ) )
= ( arctan @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arctan_double
thf(fact_2344_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_2345_divmod__BitM__2__eq,axiom,
! [M2: num] :
( ( unique8689654367752047608divmod @ int @ ( bitM @ M2 ) @ ( bit0 @ one2 ) )
= ( product_Pair @ int @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% divmod_BitM_2_eq
thf(fact_2346_dbl__inc__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] :
( ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K2 ) )
= ( numeral_numeral @ A @ ( bit1 @ K2 ) ) ) ) ).
% dbl_inc_simps(5)
thf(fact_2347_dbl__dec__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] :
( ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K2 ) )
= ( numeral_numeral @ A @ ( bitM @ K2 ) ) ) ) ).
% dbl_dec_simps(5)
thf(fact_2348_and__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( one_one @ A ) ) ) ).
% and_numerals(2)
thf(fact_2349_and__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% and_numerals(8)
thf(fact_2350_pred__numeral__simps_I2_J,axiom,
! [K2: num] :
( ( pred_numeral @ ( bit0 @ K2 ) )
= ( numeral_numeral @ nat @ ( bitM @ K2 ) ) ) ).
% pred_numeral_simps(2)
thf(fact_2351_dbl__dec__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] :
( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K2 ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K2 ) ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_2352_dbl__inc__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] :
( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K2 ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K2 ) ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_2353_and__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(5)
thf(fact_2354_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_2355_and__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(3)
thf(fact_2356_and__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
= ( one_one @ int ) ) ).
% and_minus_numerals(6)
thf(fact_2357_and__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( one_one @ int ) ) ).
% and_minus_numerals(2)
thf(fact_2358_and__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(6)
thf(fact_2359_and__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(4)
thf(fact_2360_and__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
= ( zero_zero @ int ) ) ).
% and_minus_numerals(5)
thf(fact_2361_and__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( zero_zero @ int ) ) ).
% and_minus_numerals(1)
thf(fact_2362_and__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% and_numerals(7)
thf(fact_2363_semiring__norm_I26_J,axiom,
( ( bitM @ one2 )
= one2 ) ).
% semiring_norm(26)
thf(fact_2364_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_2365_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_2366_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_2367_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_2368_even__and__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ A3 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_and_iff
thf(fact_2369_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_2370_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_2371_even__and__iff__int,axiom,
! [K2: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ K2 @ L ) )
= ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
| ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ).
% even_and_iff_int
thf(fact_2372_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X4: A] : ( plus_plus @ A @ ( plus_plus @ A @ X4 @ X4 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_2373_and__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( one_one @ A ) )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% and_one_eq
thf(fact_2374_one__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A3 )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% one_and_eq
thf(fact_2375_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_2376_odd__numeral__BitM,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [W: num] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bitM @ W ) ) ) ) ).
% odd_numeral_BitM
thf(fact_2377_even__abs__add__iff,axiom,
! [K2: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ ( abs_abs @ int @ K2 ) @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K2 @ L ) ) ) ).
% even_abs_add_iff
thf(fact_2378_even__add__abs__iff,axiom,
! [K2: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K2 @ ( abs_abs @ int @ L ) ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K2 @ L ) ) ) ).
% even_add_abs_iff
thf(fact_2379_nat__intermed__int__val,axiom,
! [M2: nat,N: nat,F3: nat > int,K2: int] :
( ! [I3: nat] :
( ( ( ord_less_eq @ nat @ M2 @ I3 )
& ( ord_less @ nat @ I3 @ N ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( suc @ I3 ) ) @ ( F3 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ int @ ( F3 @ M2 ) @ K2 )
=> ( ( ord_less_eq @ int @ K2 @ ( F3 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ M2 @ I3 )
& ( ord_less_eq @ nat @ I3 @ N )
& ( ( F3 @ I3 )
= K2 ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_2380_and__int__rec,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L2: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_2381_nat__ivt__aux,axiom,
! [N: nat,F3: nat > int,K2: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( suc @ I3 ) ) @ ( F3 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F3 @ ( zero_zero @ nat ) ) @ K2 )
=> ( ( ord_less_eq @ int @ K2 @ ( F3 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( F3 @ I3 )
= K2 ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_2382_nat0__intermed__int__val,axiom,
! [N: nat,F3: nat > int,K2: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) @ ( F3 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F3 @ ( zero_zero @ nat ) ) @ K2 )
=> ( ( ord_less_eq @ int @ K2 @ ( F3 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( F3 @ I3 )
= K2 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_2383_and__int__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( ( K3
= ( zero_zero @ int ) )
| ( L2
= ( zero_zero @ int ) ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ L2
@ ( if @ int
@ ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ K3
@ ( plus_plus @ int @ ( times_times @ int @ ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_2384_and__int_Opsimps,axiom,
! [K2: int,L: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L ) )
=> ( ( ( ( member @ int @ K2 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K2 @ L )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member @ int @ K2 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K2 @ L )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_2385_and__int_Opelims,axiom,
! [X3: int,Xa2: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X3 @ Xa2 ) )
=> ~ ( ( ( ( ( member @ int @ X3 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa2 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member @ int @ X3 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa2 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X3 @ Xa2 ) ) ) ) ) ).
% and_int.pelims
thf(fact_2386_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_2387_signed__take__bit__eq__take__bit__minus,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ K3 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_2388_mask__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: num] :
( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N ) ) ) ) ) ) ).
% mask_numeral
thf(fact_2389_num_Osize__gen_I3_J,axiom,
! [X32: num] :
( ( size_num @ ( bit1 @ X32 ) )
= ( plus_plus @ nat @ ( size_num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(3)
thf(fact_2390_of__int__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z2: int,N: num] :
( ( ( ring_1_of_int @ A @ Z2 )
= ( numeral_numeral @ A @ N ) )
= ( Z2
= ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_2391_of__int__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K2: num] :
( ( ring_1_of_int @ A @ ( numeral_numeral @ int @ K2 ) )
= ( numeral_numeral @ A @ K2 ) ) ) ).
% of_int_numeral
thf(fact_2392_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ W @ Z2 ) ) ) ).
% of_int_le_iff
thf(fact_2393_of__int__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z2: int] :
( ( ring_1_of_int @ A @ ( plus_plus @ int @ W @ Z2 ) )
= ( plus_plus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_add
thf(fact_2394_of__int__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: int,N: nat] :
( ( ring_1_of_int @ A @ ( power_power @ int @ Z2 @ N ) )
= ( power_power @ A @ ( ring_1_of_int @ A @ Z2 ) @ N ) ) ) ).
% of_int_power
thf(fact_2395_of__int__eq__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [B2: int,W: nat,X3: int] :
( ( ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W )
= ( ring_1_of_int @ A @ X3 ) )
= ( ( power_power @ int @ B2 @ W )
= X3 ) ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_2396_of__int__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X3: int,B2: int,W: nat] :
( ( ( ring_1_of_int @ A @ X3 )
= ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( X3
= ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_2397_and__nat__numerals_I3_J,axiom,
! [X3: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(3)
thf(fact_2398_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_2399_bit__numeral__Bit0__Suc__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M2 ) @ N ) ) ) ).
% bit_numeral_Bit0_Suc_iff
thf(fact_2400_bit__numeral__Bit1__Suc__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M2 ) @ N ) ) ) ).
% bit_numeral_Bit1_Suc_iff
thf(fact_2401_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_2402_of__int__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) ) ) ) ).
% of_int_le_0_iff
thf(fact_2403_of__int__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ).
% of_int_0_le_iff
thf(fact_2404_of__int__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z2: int] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ ( numeral_numeral @ int @ N ) @ Z2 ) ) ) ).
% of_int_numeral_le_iff
thf(fact_2405_of__int__le__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int,N: num] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ int @ Z2 @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_le_numeral_iff
thf(fact_2406_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_2407_and__nat__numerals_I4_J,axiom,
! [X3: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(4)
thf(fact_2408_of__int__less__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int,N: num] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ int @ Z2 @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_less_numeral_iff
thf(fact_2409_of__int__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z2: int] :
( ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less @ int @ ( numeral_numeral @ int @ N ) @ Z2 ) ) ) ).
% of_int_numeral_less_iff
thf(fact_2410_of__int__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ int @ Z2 @ ( one_one @ int ) ) ) ) ).
% of_int_le_1_iff
thf(fact_2411_of__int__1__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% of_int_1_le_iff
thf(fact_2412_of__int__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X3: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_2413_numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X3: num,N: nat,Y: int] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N )
= Y ) ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_2414_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: int,B2: int,W: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( ord_less_eq @ int @ X3 @ ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_2415_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W: nat,X3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) @ ( ring_1_of_int @ A @ X3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W ) @ X3 ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_2416_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W: nat,X3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) @ ( ring_1_of_int @ A @ X3 ) )
= ( ord_less @ int @ ( power_power @ int @ B2 @ W ) @ X3 ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_2417_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: int,B2: int,W: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X3 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( ord_less @ int @ X3 @ ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_2418_bit__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_numeral_simps(2)
thf(fact_2419_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ N ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_2420_bit__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_numeral_simps(3)
thf(fact_2421_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ N ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_2422_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_2423_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_2424_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_2425_bit__minus__numeral__int_I1_J,axiom,
! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% bit_minus_numeral_int(1)
thf(fact_2426_bit__minus__numeral__int_I2_J,axiom,
! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) @ ( numeral_numeral @ nat @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_minus_numeral_int(2)
thf(fact_2427_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_2428_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X3: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_2429_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: num,N: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) @ A3 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_2430_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X3: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_2431_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: num,N: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) @ A3 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_2432_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X3: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N ) )
= ( Y
= ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N ) ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_2433_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X3: num,N: nat,Y: int] :
( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N )
= Y ) ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_2434_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X3: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_2435_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: num,N: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N ) @ A3 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_2436_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: num,N: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N ) @ A3 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_2437_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X3: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_2438_bit__numeral__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M2 ) @ N )
= ( bit_se5641148757651400278ts_bit @ nat @ ( numeral_numeral @ nat @ M2 ) @ N ) ) ) ).
% bit_numeral_iff
thf(fact_2439_bit__disjunctive__add__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,B2: A,N: nat] :
( ! [N2: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N2 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
| ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ) ).
% bit_disjunctive_add_iff
thf(fact_2440_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% less_eq_mask
thf(fact_2441_not__bit__1__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N ) ) ) ).
% not_bit_1_Suc
thf(fact_2442_bit__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: num] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% bit_numeral_simps(1)
thf(fact_2443_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_2444_of__int__nonneg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_nonneg
thf(fact_2445_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X3 )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ) ).
% of_int_leD
thf(fact_2446_of__int__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K2: num] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ K2 ) ) ) ) ).
% of_int_neg_numeral
thf(fact_2447_bit__concat__bit__iff,axiom,
! [M2: nat,K2: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M2 @ K2 @ L ) @ N )
= ( ( ( ord_less @ nat @ N @ M2 )
& ( bit_se5641148757651400278ts_bit @ int @ K2 @ N ) )
| ( ( ord_less_eq @ nat @ M2 @ N )
& ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_2448_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_2449_bit__Suc,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) ) ).
% bit_Suc
thf(fact_2450_stable__imp__bit__iff__odd,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% stable_imp_bit_iff_odd
thf(fact_2451_bit__iff__idd__imp__stable,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ! [N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N2 )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
=> ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 ) ) ) ).
% bit_iff_idd_imp_stable
thf(fact_2452_int__bit__bound,axiom,
! [K2: int] :
~ ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ N2 @ M3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K2 @ M3 )
= ( bit_se5641148757651400278ts_bit @ int @ K2 @ N2 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N2 ) ) ) ) ) ).
% int_bit_bound
thf(fact_2453_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_2454_bit__iff__odd,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A8: A,N3: nat] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A8 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ) ).
% bit_iff_odd
thf(fact_2455_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_2456_Suc__mask__eq__exp,axiom,
! [N: nat] :
( ( suc @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% Suc_mask_eq_exp
thf(fact_2457_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_2458_even__of__int__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K2: int] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ K2 ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 ) ) ) ).
% even_of_int_iff
thf(fact_2459_bit__int__def,axiom,
( ( bit_se5641148757651400278ts_bit @ int )
= ( ^ [K3: int,N3: nat] :
~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% bit_int_def
thf(fact_2460_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_2461_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_2462_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_2463_mask__nat__def,axiom,
( ( bit_se2239418461657761734s_mask @ nat )
= ( ^ [N3: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) ) ).
% mask_nat_def
thf(fact_2464_mask__half__int,axiom,
! [N: nat] :
( ( divide_divide @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_se2239418461657761734s_mask @ int @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% mask_half_int
thf(fact_2465_mask__int__def,axiom,
( ( bit_se2239418461657761734s_mask @ int )
= ( ^ [N3: nat] : ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ int ) ) ) ) ).
% mask_int_def
thf(fact_2466_set__bit__eq,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N3: nat,K3: int] :
( plus_plus @ int @ K3
@ ( times_times @ int
@ ( zero_neq_one_of_bool @ int
@ ~ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) )
@ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% set_bit_eq
thf(fact_2467_unset__bit__eq,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ K3 @ ( times_times @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_2468_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: 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 ) ) @ B2 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_2469_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A8: A,N3: nat] :
( ( ( N3
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A8 ) )
& ( ( N3
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_2470_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A )
= ( ^ [N3: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ A ) ) ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_2471_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M5: nat,N3: nat] :
( if @ nat
@ ( ( M5
= ( zero_zero @ nat ) )
| ( N3
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_2472_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M5: nat,N3: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_2473_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [K: int,L4: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L4 ) )
=> ( ( ~ ( ( member @ int @ K @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L4 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K @ L4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_2474_take__bit__Suc__from__most,axiom,
! [N: nat,K2: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K2 )
= ( plus_plus @ int @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) ) ).
% take_bit_Suc_from_most
thf(fact_2475_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K2: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K2 )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( plus_plus @ int @ K2 @ ( one_one @ int ) ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_2476_num_Osize__gen_I2_J,axiom,
! [X2: num] :
( ( size_num @ ( bit0 @ X2 ) )
= ( plus_plus @ nat @ ( size_num @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(2)
thf(fact_2477_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [X5: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X5 ) @ X3 )
& ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X5 @ ( one_one @ int ) ) ) )
& ! [Y6: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y6 ) @ X3 )
& ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y6 @ ( one_one @ int ) ) ) ) )
=> ( Y6 = X5 ) ) ) ) ).
% floor_exists1
thf(fact_2478_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X3 )
& ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_2479_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [I3: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I3 @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I3 @ J2 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) )
=> ( P @ I3 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_2480_tanh__real__altdef,axiom,
( ( tanh @ real )
= ( ^ [X4: real] : ( divide_divide @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X4 ) ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X4 ) ) ) ) ) ) ).
% tanh_real_altdef
thf(fact_2481_power__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K2: num,L: num] :
( ( power_power @ A @ ( numeral_numeral @ A @ K2 ) @ ( numeral_numeral @ nat @ L ) )
= ( numeral_numeral @ A @ ( pow @ K2 @ L ) ) ) ) ).
% power_numeral
thf(fact_2482_or__int__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
| ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
@ ( uminus_uminus @ int @ ( one_one @ int ) )
@ ( if @ int
@ ( K3
= ( zero_zero @ int ) )
@ L2
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K3
@ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_2483_or__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% or_numerals(2)
thf(fact_2484_or__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X3 ) ) ) ) ).
% or_numerals(8)
thf(fact_2485_or__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% or_numerals(3)
thf(fact_2486_or__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% or_numerals(1)
thf(fact_2487_or__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X3 ) ) ) ) ).
% or_numerals(5)
thf(fact_2488_or__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(6)
thf(fact_2489_or__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(2)
thf(fact_2490_or__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(7)
thf(fact_2491_or__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(6)
thf(fact_2492_or__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(4)
thf(fact_2493_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_2494_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_2495_disjunctive__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: A] :
( ! [N2: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N2 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 ) )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( bit_se1065995026697491101ons_or @ A @ A3 @ B2 ) ) ) ) ).
% disjunctive_add
thf(fact_2496_exp__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A,Y: A] :
( ( ( times_times @ A @ X3 @ Y )
= ( times_times @ A @ Y @ X3 ) )
=> ( ( exp @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( times_times @ A @ ( exp @ A @ X3 ) @ ( exp @ A @ Y ) ) ) ) ) ).
% exp_add_commuting
thf(fact_2497_mult__exp__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( times_times @ A @ ( exp @ A @ X3 ) @ ( exp @ A @ Y ) )
= ( exp @ A @ ( plus_plus @ A @ X3 @ Y ) ) ) ) ).
% mult_exp_exp
thf(fact_2498_pow_Osimps_I1_J,axiom,
! [X3: num] :
( ( pow @ X3 @ one2 )
= X3 ) ).
% pow.simps(1)
thf(fact_2499_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_2500_even__or__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ A3 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_or_iff
thf(fact_2501_exp__le,axiom,
ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ).
% exp_le
thf(fact_2502_tanh__altdef,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A )
= ( ^ [X4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ X4 ) @ ( exp @ A @ ( uminus_uminus @ A @ X4 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X4 ) @ ( exp @ A @ ( uminus_uminus @ A @ X4 ) ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_2503_bit__nat__def,axiom,
( ( bit_se5641148757651400278ts_bit @ nat )
= ( ^ [M5: nat,N3: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ M5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% bit_nat_def
thf(fact_2504_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_2505_mask__Suc__exp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N ) )
= ( bit_se1065995026697491101ons_or @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% mask_Suc_exp
thf(fact_2506_exp__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z2: A] :
( ( exp @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) )
= ( power_power @ A @ ( exp @ A @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% exp_double
thf(fact_2507_one__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A3 )
= ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% one_or_eq
thf(fact_2508_or__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( one_one @ A ) )
= ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% or_one_eq
thf(fact_2509_mask__Suc__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N ) )
= ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ).
% mask_Suc_double
thf(fact_2510_OR__upper,axiom,
! [X3: int,N: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less @ int @ X3 @ ( 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 @ X3 @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_2511_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_2512_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [Z3: int] : ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_le_of_int
thf(fact_2513_exp__bound,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X3 ) @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% exp_bound
thf(fact_2514_or__int__rec,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K3: int,L2: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
| ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_2515_real__exp__bound__lemma,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X3 ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) ) ) ) ) ).
% real_exp_bound_lemma
thf(fact_2516_exp__lower__Taylor__quadratic,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) @ ( divide_divide @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ X3 ) ) ) ).
% exp_lower_Taylor_quadratic
thf(fact_2517_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y ) )
=> ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y ) @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X3 )
= Y ) ) ) ) ).
% round_unique
thf(fact_2518_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,N: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X3 )
= N ) ) ) ).
% round_unique'
thf(fact_2519_of__int__round__abs__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) @ X3 ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_int_round_abs_le
thf(fact_2520_or__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(5)
thf(fact_2521_or__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(1)
thf(fact_2522_of__int__round__gt,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less @ A @ ( minus_minus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) ) ) ).
% of_int_round_gt
thf(fact_2523_round__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ int @ N ) ) ) ).
% round_numeral
thf(fact_2524_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_2525_or__nat__numerals_I4_J,axiom,
! [X3: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) ) ).
% or_nat_numerals(4)
thf(fact_2526_or__nat__numerals_I3_J,axiom,
! [X3: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) ) ).
% or_nat_numerals(3)
thf(fact_2527_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_2528_round__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ).
% round_neg_numeral
thf(fact_2529_or__minus__numerals_I4_J,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M2 @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(4)
thf(fact_2530_or__minus__numerals_I8_J,axiom,
! [N: num,M2: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M2 ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M2 @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(8)
thf(fact_2531_or__minus__numerals_I7_J,axiom,
! [N: num,M2: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M2 ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(7)
thf(fact_2532_or__minus__numerals_I3_J,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(3)
thf(fact_2533_or__not__num__neg_Osimps_I1_J,axiom,
( ( bit_or_not_num_neg @ one2 @ one2 )
= one2 ) ).
% or_not_num_neg.simps(1)
thf(fact_2534_or__not__num__neg_Osimps_I4_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one2 )
= ( bit0 @ one2 ) ) ).
% or_not_num_neg.simps(4)
thf(fact_2535_or__not__num__neg_Osimps_I6_J,axiom,
! [N: num,M2: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M2 ) )
= ( bit0 @ ( bit_or_not_num_neg @ N @ M2 ) ) ) ).
% or_not_num_neg.simps(6)
thf(fact_2536_or__not__num__neg_Osimps_I3_J,axiom,
! [M2: num] :
( ( bit_or_not_num_neg @ one2 @ ( bit1 @ M2 ) )
= ( bit1 @ M2 ) ) ).
% or_not_num_neg.simps(3)
thf(fact_2537_or__not__num__neg_Osimps_I7_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one2 )
= one2 ) ).
% or_not_num_neg.simps(7)
thf(fact_2538_or__not__num__neg_Osimps_I5_J,axiom,
! [N: num,M2: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M2 ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M2 ) ) ) ).
% or_not_num_neg.simps(5)
thf(fact_2539_round__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X3 ) @ ( archimedean_round @ A @ Y ) ) ) ) ).
% round_mono
thf(fact_2540_or__not__num__neg_Osimps_I2_J,axiom,
! [M2: num] :
( ( bit_or_not_num_neg @ one2 @ ( bit0 @ M2 ) )
= ( bit1 @ M2 ) ) ).
% or_not_num_neg.simps(2)
thf(fact_2541_or__not__num__neg_Osimps_I8_J,axiom,
! [N: num,M2: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M2 ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M2 ) ) ) ).
% or_not_num_neg.simps(8)
thf(fact_2542_or__not__num__neg_Oelims,axiom,
! [X3: num,Xa2: num,Y: num] :
( ( ( bit_or_not_num_neg @ X3 @ Xa2 )
= Y )
=> ( ( ( X3 = one2 )
=> ( ( Xa2 = one2 )
=> ( Y != one2 ) ) )
=> ( ( ( X3 = one2 )
=> ! [M: num] :
( ( Xa2
= ( bit0 @ M ) )
=> ( Y
!= ( bit1 @ M ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [M: num] :
( ( Xa2
= ( bit1 @ M ) )
=> ( Y
!= ( bit1 @ M ) ) ) )
=> ( ( ? [N2: num] :
( X3
= ( bit0 @ N2 ) )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( bit0 @ one2 ) ) ) )
=> ( ! [N2: num] :
( ( X3
= ( bit0 @ N2 ) )
=> ! [M: num] :
( ( Xa2
= ( bit0 @ M ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) )
=> ( ! [N2: num] :
( ( X3
= ( bit0 @ N2 ) )
=> ! [M: num] :
( ( Xa2
= ( bit1 @ M ) )
=> ( Y
!= ( bit0 @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) )
=> ( ( ? [N2: num] :
( X3
= ( bit1 @ N2 ) )
=> ( ( Xa2 = one2 )
=> ( Y != one2 ) ) )
=> ( ! [N2: num] :
( ( X3
= ( bit1 @ N2 ) )
=> ! [M: num] :
( ( Xa2
= ( bit0 @ M ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) )
=> ~ ! [N2: num] :
( ( X3
= ( bit1 @ N2 ) )
=> ! [M: num] :
( ( Xa2
= ( bit1 @ M ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.elims
thf(fact_2543_round__diff__minimal,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: A,M2: int] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ Z2 @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ Z2 ) ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ Z2 @ ( ring_1_of_int @ A @ M2 ) ) ) ) ) ).
% round_diff_minimal
thf(fact_2544_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_2545_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_2546_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M5: nat,N3: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_2547_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M5: nat,N3: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N3
@ ( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ M5
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_2548_of__int__round__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% of_int_round_le
thf(fact_2549_of__int__round__ge,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) ) ) ).
% of_int_round_ge
thf(fact_2550_log__base__10__eq1,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X3 )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( ln_ln @ real @ X3 ) ) ) ) ).
% log_base_10_eq1
thf(fact_2551_arctan__half,axiom,
( arctan
= ( ^ [X4: real] : ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ ( divide_divide @ real @ X4 @ ( plus_plus @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% arctan_half
thf(fact_2552_log__base__10__eq2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X3 )
= ( times_times @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X3 ) ) ) ) ).
% log_base_10_eq2
thf(fact_2553_machin,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% machin
thf(fact_2554_machin__Euler,axiom,
( ( plus_plus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% machin_Euler
thf(fact_2555_sqrt__sum__squares__half__less,axiom,
! [X3: real,U: real,Y: real] :
( ( ord_less @ real @ X3 @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_2556_real__sqrt__four,axiom,
( ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% real_sqrt_four
thf(fact_2557_real__sqrt__abs,axiom,
! [X3: real] :
( ( sqrt @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( abs_abs @ real @ X3 ) ) ).
% real_sqrt_abs
thf(fact_2558_real__sqrt__pow2,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( power_power @ real @ ( sqrt @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X3 ) ) ).
% real_sqrt_pow2
thf(fact_2559_real__sqrt__pow2__iff,axiom,
! [X3: real] :
( ( ( power_power @ real @ ( sqrt @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X3 )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% real_sqrt_pow2_iff
thf(fact_2560_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X3: real,Y: real,Xa2: real,Ya: real] :
( ( power_power @ real @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_squared_eq
thf(fact_2561_real__sqrt__power,axiom,
! [X3: real,K2: nat] :
( ( sqrt @ ( power_power @ real @ X3 @ K2 ) )
= ( power_power @ real @ ( sqrt @ X3 ) @ K2 ) ) ).
% real_sqrt_power
thf(fact_2562_sqrt2__less__2,axiom,
ord_less @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% sqrt2_less_2
thf(fact_2563_pi__less__4,axiom,
ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ).
% pi_less_4
thf(fact_2564_pi__ge__two,axiom,
ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ).
% pi_ge_two
thf(fact_2565_pi__half__neq__two,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% pi_half_neq_two
thf(fact_2566_real__less__rsqrt,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ord_less @ real @ X3 @ ( sqrt @ Y ) ) ) ).
% real_less_rsqrt
thf(fact_2567_sqrt__le__D,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( sqrt @ X3 ) @ Y )
=> ( ord_less_eq @ real @ X3 @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% sqrt_le_D
thf(fact_2568_real__le__rsqrt,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ord_less_eq @ real @ X3 @ ( sqrt @ Y ) ) ) ).
% real_le_rsqrt
thf(fact_2569_pi__half__neq__zero,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% pi_half_neq_zero
thf(fact_2570_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_2571_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_2572_real__le__lsqrt,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ X3 @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sqrt @ X3 ) @ Y ) ) ) ) ).
% real_le_lsqrt
thf(fact_2573_real__sqrt__unique,axiom,
! [Y: real,X3: real] :
( ( ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( sqrt @ X3 )
= Y ) ) ) ).
% real_sqrt_unique
thf(fact_2574_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_2575_real__sqrt__sum__squares__eq__cancel,axiom,
! [X3: real,Y: real] :
( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= X3 )
=> ( Y
= ( zero_zero @ real ) ) ) ).
% real_sqrt_sum_squares_eq_cancel
thf(fact_2576_real__sqrt__sum__squares__eq__cancel2,axiom,
! [X3: real,Y: real] :
( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= Y )
=> ( X3
= ( zero_zero @ real ) ) ) ).
% real_sqrt_sum_squares_eq_cancel2
thf(fact_2577_real__sqrt__sum__squares__ge1,axiom,
! [X3: real,Y: real] : ( ord_less_eq @ real @ X3 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge1
thf(fact_2578_real__sqrt__sum__squares__ge2,axiom,
! [Y: real,X3: real] : ( ord_less_eq @ real @ Y @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge2
thf(fact_2579_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A3: real,C3: real,B2: real,D3: 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 @ B2 @ D3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ C3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_triangle_ineq
thf(fact_2580_sqrt__ge__absD,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( sqrt @ Y ) )
=> ( ord_less_eq @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) ).
% sqrt_ge_absD
thf(fact_2581_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_2582_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_2583_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_2584_arctan__ubound,axiom,
! [Y: real] : ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arctan_ubound
thf(fact_2585_arctan__one,axiom,
( ( arctan @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_one
thf(fact_2586_real__less__lsqrt,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X3 @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sqrt @ X3 ) @ Y ) ) ) ) ).
% real_less_lsqrt
thf(fact_2587_sqrt__sum__squares__le__sum,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ X3 @ Y ) ) ) ) ).
% sqrt_sum_squares_le_sum
thf(fact_2588_sqrt__even__pow2,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( sqrt @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sqrt_even_pow2
thf(fact_2589_real__sqrt__ge__abs1,axiom,
! [X3: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs1
thf(fact_2590_real__sqrt__ge__abs2,axiom,
! [Y: real,X3: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs2
thf(fact_2591_sqrt__sum__squares__le__sum__abs,axiom,
! [X3: real,Y: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( abs_abs @ real @ X3 ) @ ( abs_abs @ real @ Y ) ) ) ).
% sqrt_sum_squares_le_sum_abs
thf(fact_2592_ln__sqrt,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ln_ln @ real @ ( sqrt @ X3 ) )
= ( divide_divide @ real @ ( ln_ln @ real @ X3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% ln_sqrt
thf(fact_2593_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_2594_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_2595_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_2596_arsinh__real__aux,axiom,
! [X3: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X3 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% arsinh_real_aux
thf(fact_2597_real__sqrt__power__even,axiom,
! [N: nat,X3: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( power_power @ real @ ( sqrt @ X3 ) @ N )
= ( power_power @ real @ X3 @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_power_even
thf(fact_2598_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X3: real,Y: real,Xa2: real,Ya: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_ge_zero
thf(fact_2599_arith__geo__mean__sqrt,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( times_times @ real @ X3 @ Y ) ) @ ( divide_divide @ real @ ( plus_plus @ real @ X3 @ Y ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arith_geo_mean_sqrt
thf(fact_2600_cos__x__y__le__one,axiom,
! [X3: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( divide_divide @ real @ X3 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( one_one @ real ) ) ).
% cos_x_y_le_one
thf(fact_2601_real__sqrt__sum__squares__less,axiom,
! [X3: real,U: real,Y: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_2602_arcosh__real__def,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 )
=> ( ( arcosh @ real @ X3 )
= ( ln_ln @ real @ ( plus_plus @ real @ X3 @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arcosh_real_def
thf(fact_2603_arsinh__real__def,axiom,
( ( arsinh @ real )
= ( ^ [X4: real] : ( ln_ln @ real @ ( plus_plus @ real @ X4 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arsinh_real_def
thf(fact_2604_cot__less__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cot @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% cot_less_zero
thf(fact_2605_cot__periodic,axiom,
! [X3: real] :
( ( cot @ real @ ( plus_plus @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( cot @ real @ X3 ) ) ).
% cot_periodic
thf(fact_2606_sin__3over2__pi,axiom,
( ( sin @ real @ ( times_times @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).
% sin_3over2_pi
thf(fact_2607_arcsin__minus__1,axiom,
( ( arcsin @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% arcsin_minus_1
thf(fact_2608_sin__int__2pin,axiom,
! [N: int] :
( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( ring_1_of_int @ real @ N ) ) )
= ( zero_zero @ real ) ) ).
% sin_int_2pin
thf(fact_2609_sin__two__pi,axiom,
( ( sin @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_two_pi
thf(fact_2610_sin__pi__half,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ).
% sin_pi_half
thf(fact_2611_sin__periodic,axiom,
! [X3: real] :
( ( sin @ real @ ( plus_plus @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( sin @ real @ X3 ) ) ).
% sin_periodic
thf(fact_2612_arcsin__1,axiom,
( ( arcsin @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arcsin_1
thf(fact_2613_sin__2pi__minus,axiom,
! [X3: real] :
( ( sin @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X3 ) )
= ( uminus_uminus @ real @ ( sin @ real @ X3 ) ) ) ).
% sin_2pi_minus
thf(fact_2614_arcsin__sin,axiom,
! [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 ) ) ) )
=> ( ( arcsin @ ( sin @ real @ X3 ) )
= X3 ) ) ) ).
% arcsin_sin
thf(fact_2615_le__arcsin__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( 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 @ X3 ) )
= ( ord_less_eq @ real @ ( sin @ real @ Y ) @ X3 ) ) ) ) ) ) ).
% le_arcsin_iff
thf(fact_2616_arcsin__le__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( 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 @ X3 ) @ Y )
= ( ord_less_eq @ real @ X3 @ ( sin @ real @ Y ) ) ) ) ) ) ) ).
% arcsin_le_iff
thf(fact_2617_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_2618_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_2619_sin__gt__zero__02,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) ) ) ) ).
% sin_gt_zero_02
thf(fact_2620_sin__45,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% sin_45
thf(fact_2621_sin__gt__zero2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) ) ) ) ).
% sin_gt_zero2
thf(fact_2622_sin__lt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ pi @ X3 )
=> ( ( ord_less @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less @ real @ ( sin @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% sin_lt_zero
thf(fact_2623_sin__30,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% sin_30
thf(fact_2624_sin__monotone__2pi__le,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sin @ real @ Y ) @ ( sin @ real @ X3 ) ) ) ) ) ).
% sin_monotone_2pi_le
thf(fact_2625_sin__mono__le__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( sin @ real @ X3 ) @ ( sin @ real @ Y ) )
= ( ord_less_eq @ real @ X3 @ Y ) ) ) ) ) ) ).
% sin_mono_le_eq
thf(fact_2626_sin__inj__pi,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( sin @ real @ X3 )
= ( sin @ real @ Y ) )
=> ( X3 = Y ) ) ) ) ) ) ).
% sin_inj_pi
thf(fact_2627_sin__60,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% sin_60
thf(fact_2628_sin__le__zero,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ pi @ X3 )
=> ( ( ord_less @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less_eq @ real @ ( sin @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% sin_le_zero
thf(fact_2629_sin__less__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( sin @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% sin_less_zero
thf(fact_2630_sin__mono__less__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( sin @ real @ X3 ) @ ( sin @ real @ Y ) )
= ( ord_less @ real @ X3 @ Y ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_2631_sin__monotone__2pi,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sin @ real @ Y ) @ ( sin @ real @ X3 ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_2632_sin__total,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ? [X5: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X5 )
& ( ord_less_eq @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X5 )
= Y )
& ! [Y6: real] :
( ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y6 )
& ( ord_less_eq @ real @ Y6 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ Y6 )
= Y ) )
=> ( Y6 = X5 ) ) ) ) ) ).
% sin_total
thf(fact_2633_sin__zero__iff__int,axiom,
! [X3: real] :
( ( ( sin @ real @ X3 )
= ( zero_zero @ real ) )
= ( ? [I4: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I4 )
& ( X3
= ( times_times @ real @ ( ring_1_of_int @ real @ I4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_iff_int
thf(fact_2634_sin__arctan,axiom,
! [X3: real] :
( ( sin @ real @ ( arctan @ X3 ) )
= ( divide_divide @ real @ X3 @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arctan
thf(fact_2635_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_2636_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_2637_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_2638_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_2639_cot__gt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cot @ real @ X3 ) ) ) ) ).
% cot_gt_zero
thf(fact_2640_sin__cos__npi,axiom,
! [N: nat] :
( ( sin @ real @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) ) ).
% sin_cos_npi
thf(fact_2641_sincos__total__2pi,axiom,
! [X3: real,Y: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ~ ! [T6: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T6 )
=> ( ( ord_less @ real @ T6 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ( X3
= ( cos @ real @ T6 ) )
=> ( Y
!= ( sin @ real @ T6 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_2642_sin__tan,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( sin @ real @ X3 )
= ( divide_divide @ real @ ( tan @ real @ X3 ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_tan
thf(fact_2643_sincos__total__2pi__le,axiom,
! [X3: real,Y: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T6: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less_eq @ real @ T6 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
& ( X3
= ( cos @ real @ T6 ) )
& ( Y
= ( sin @ real @ T6 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_2644_sincos__total__pi__half,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X3 @ ( 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 ) ) ) )
& ( X3
= ( cos @ real @ T6 ) )
& ( Y
= ( sin @ real @ T6 ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_2645_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_2646_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M2: nat,N: nat] :
( ( ( semiring_1_of_nat @ A @ M2 )
= ( semiring_1_of_nat @ A @ N ) )
= ( M2 = N ) ) ) ).
% of_nat_eq_iff
thf(fact_2647_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_2648_of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M2: nat] :
( ( ( semiring_1_of_nat @ A @ M2 )
= ( zero_zero @ A ) )
= ( M2
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_eq_0_iff
thf(fact_2649_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_2650_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_2651_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ).
% of_nat_less_iff
thf(fact_2652_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_2653_of__nat__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% of_nat_le_iff
thf(fact_2654_of__nat__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_add
thf(fact_2655_of__nat__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M2 @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mult
thf(fact_2656_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_2657_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_2658_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_2659_of__nat__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M2 @ N ) )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ M2 ) @ N ) ) ) ).
% of_nat_power
thf(fact_2660_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [B2: nat,W: nat,X3: nat] :
( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W )
= ( semiring_1_of_nat @ A @ X3 ) )
= ( ( power_power @ nat @ B2 @ W )
= X3 ) ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_2661_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X3: nat,B2: nat,W: nat] :
( ( ( semiring_1_of_nat @ A @ X3 )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
= ( X3
= ( power_power @ nat @ B2 @ W ) ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_2662_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_2663_of__nat__le__0__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( zero_zero @ A ) )
= ( M2
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_le_0_iff
thf(fact_2664_of__nat__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ M2 ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ).
% of_nat_Suc
thf(fact_2665_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_2666_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W: nat,X3: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) @ ( semiring_1_of_nat @ A @ X3 ) )
= ( ord_less @ nat @ ( power_power @ nat @ B2 @ W ) @ X3 ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_2667_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,B2: nat,W: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
= ( ord_less @ nat @ X3 @ ( power_power @ nat @ B2 @ W ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_2668_numeral__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X3: num,N: nat,Y: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N )
= ( semiring_1_of_nat @ A @ Y ) )
= ( ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N )
= Y ) ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_2669_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [Y: nat,X3: num,N: nat] :
( ( ( semiring_1_of_nat @ A @ Y )
= ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) )
= ( Y
= ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N ) ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_2670_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W: nat,X3: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) @ ( semiring_1_of_nat @ A @ X3 ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W ) @ X3 ) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_2671_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,B2: nat,W: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
= ( ord_less_eq @ nat @ X3 @ ( power_power @ nat @ B2 @ W ) ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_2672_sin__cos__squared__add3,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ X3 ) ) @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ X3 ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add3
thf(fact_2673_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W: num] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( numeral_numeral @ real @ W ) )
= ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ W ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_2674_numeral__less__real__of__nat__iff,axiom,
! [W: num,N: nat] :
( ( ord_less @ real @ ( numeral_numeral @ real @ W ) @ ( semiring_1_of_nat @ real @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ W ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_2675_numeral__le__real__of__nat__iff,axiom,
! [N: num,M2: nat] :
( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N ) @ ( semiring_1_of_nat @ real @ M2 ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N ) @ M2 ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_2676_tan__periodic__n,axiom,
! [X3: real,N: num] :
( ( tan @ real @ ( plus_plus @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ N ) @ pi ) ) )
= ( tan @ real @ X3 ) ) ).
% tan_periodic_n
thf(fact_2677_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X3 ) @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_2678_log__pow__cancel,axiom,
! [A3: real,B2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log @ A3 @ ( power_power @ real @ A3 @ B2 ) )
= ( semiring_1_of_nat @ real @ B2 ) ) ) ) ).
% log_pow_cancel
thf(fact_2679_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X3: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X3 ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X3 ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_2680_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,I: num,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_2681_even__of__nat,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ N ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% even_of_nat
thf(fact_2682_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X3: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X3 ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X3 ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_2683_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,I: num,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less_eq @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_2684_cos__pi__half,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ).
% cos_pi_half
thf(fact_2685_cos__two__pi,axiom,
( ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ real ) ) ).
% cos_two_pi
thf(fact_2686_cos__periodic,axiom,
! [X3: real] :
( ( cos @ real @ ( plus_plus @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( cos @ real @ X3 ) ) ).
% cos_periodic
thf(fact_2687_arccos__0,axiom,
( ( arccos @ ( zero_zero @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arccos_0
thf(fact_2688_cos__2pi__minus,axiom,
! [X3: real] :
( ( cos @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X3 ) )
= ( cos @ real @ X3 ) ) ).
% cos_2pi_minus
thf(fact_2689_tan__periodic,axiom,
! [X3: real] :
( ( tan @ real @ ( plus_plus @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( tan @ real @ X3 ) ) ).
% tan_periodic
thf(fact_2690_cos__npi2,axiom,
! [N: nat] :
( ( cos @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) )
= ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) ) ).
% cos_npi2
thf(fact_2691_cos__npi,axiom,
! [N: nat] :
( ( cos @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) ) ).
% cos_npi
thf(fact_2692_sin__cos__squared__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add
thf(fact_2693_sin__cos__squared__add2,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add2
thf(fact_2694_sin__2npi,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_2npi
thf(fact_2695_cos__2npi,axiom,
! [N: nat] :
( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) )
= ( one_one @ real ) ) ).
% cos_2npi
thf(fact_2696_cos__int__2pin,axiom,
! [N: int] :
( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( ring_1_of_int @ real @ N ) ) )
= ( one_one @ real ) ) ).
% cos_int_2pin
thf(fact_2697_cos__3over2__pi,axiom,
( ( cos @ real @ ( times_times @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% cos_3over2_pi
thf(fact_2698_cos__npi__int,axiom,
! [N: int] :
( ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( cos @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N ) ) )
= ( one_one @ real ) ) )
& ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( cos @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N ) ) )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ).
% cos_npi_int
thf(fact_2699_cos__pi__eq__zero,axiom,
! [M2: nat] :
( ( cos @ real @ ( divide_divide @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ).
% cos_pi_eq_zero
thf(fact_2700_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [N2: nat] : ( ord_less_eq @ A @ X3 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% real_arch_simple
thf(fact_2701_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X3: nat,Y: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X3 ) @ Y )
= ( times_times @ A @ Y @ ( semiring_1_of_nat @ A @ X3 ) ) ) ) ).
% mult_of_nat_commute
thf(fact_2702_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_2703_of__nat__less__0__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat] :
~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( zero_zero @ A ) ) ) ).
% of_nat_less_0_iff
thf(fact_2704_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_2705_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% of_nat_less_imp_less
thf(fact_2706_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% less_imp_of_nat_less
thf(fact_2707_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_2708_of__nat__max,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X3 @ Y ) )
= ( ord_max @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_max
thf(fact_2709_add__tan__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) )
= ( divide_divide @ A @ ( sin @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_2710_sin__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( sin @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( sin @ A @ Y ) ) ) ) ) ).
% sin_add
thf(fact_2711_lemma__tan__add1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) )
= ( divide_divide @ A @ ( cos @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_2712_tan__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( minus_minus @ A @ X3 @ Y ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( minus_minus @ A @ X3 @ Y ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_2713_tan__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( plus_plus @ A @ X3 @ Y ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_2714_of__nat__diff,axiom,
! [A: $tType] :
( ( semiring_1_cancel @ A )
=> ! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% of_nat_diff
thf(fact_2715_exp__of__nat2__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,N: nat] :
( ( exp @ A @ ( times_times @ A @ X3 @ ( semiring_1_of_nat @ A @ N ) ) )
= ( power_power @ A @ ( exp @ A @ X3 ) @ N ) ) ) ).
% exp_of_nat2_mult
thf(fact_2716_exp__of__nat__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X3: A] :
( ( exp @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ X3 ) )
= ( power_power @ A @ ( exp @ A @ X3 ) @ N ) ) ) ).
% exp_of_nat_mult
thf(fact_2717_cos__one__2pi,axiom,
! [X3: real] :
( ( ( cos @ real @ X3 )
= ( one_one @ real ) )
= ( ? [X4: nat] :
( X3
= ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X4 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
| ? [X4: nat] :
( X3
= ( uminus_uminus @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X4 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ) ).
% cos_one_2pi
thf(fact_2718_arccos__cos__eq__abs__2pi,axiom,
! [Theta: real] :
~ ! [K: int] :
( ( arccos @ ( cos @ real @ Theta ) )
!= ( abs_abs @ real @ ( minus_minus @ real @ Theta @ ( times_times @ real @ ( ring_1_of_int @ real @ K ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) ) ) ) ).
% arccos_cos_eq_abs_2pi
thf(fact_2719_cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( cos @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( minus_minus @ A @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ Y ) ) ) ) ) ).
% cos_add
thf(fact_2720_cos__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( cos @ A @ ( minus_minus @ A @ X3 @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ Y ) ) ) ) ) ).
% cos_diff
thf(fact_2721_cos__two__neq__zero,axiom,
( ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% cos_two_neq_zero
thf(fact_2722_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_2723_tan__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X4: A] : ( divide_divide @ A @ ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X4 ) ) @ ( plus_plus @ A @ ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X4 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% tan_half
thf(fact_2724_nat__le__real__less,axiom,
( ( ord_less_eq @ nat )
= ( ^ [N3: nat,M5: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M5 ) @ ( one_one @ real ) ) ) ) ) ).
% nat_le_real_less
thf(fact_2725_less__log__of__power,axiom,
! [B2: real,N: nat,M2: real] :
( ( ord_less @ real @ ( power_power @ real @ B2 @ N ) @ M2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ M2 ) ) ) ) ).
% less_log_of_power
thf(fact_2726_log__of__power__eq,axiom,
! [M2: nat,B2: real,N: nat] :
( ( ( semiring_1_of_nat @ real @ M2 )
= ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( semiring_1_of_nat @ real @ N )
= ( log @ B2 @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ) ).
% log_of_power_eq
thf(fact_2727_sin__expansion__lemma,axiom,
! [X3: real,M2: nat] :
( ( sin @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M2 ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( cos @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M2 ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_expansion_lemma
thf(fact_2728_sin__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ X3 ) ) @ ( cos @ A @ X3 ) ) ) ) ).
% sin_double
thf(fact_2729_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
=> ~ ! [N2: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ E3 ) ) ) ).
% nat_approx_posE
thf(fact_2730_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_2731_cos__zero__lemma,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( cos @ real @ X3 )
= ( zero_zero @ real ) )
=> ? [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X3
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_2732_cos__two__less__zero,axiom,
ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_less_zero
thf(fact_2733_cos__is__zero,axiom,
? [X5: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X5 )
& ( ord_less_eq @ real @ X5 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X5 )
= ( zero_zero @ real ) )
& ! [Y6: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y6 )
& ( ord_less_eq @ real @ Y6 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ Y6 )
= ( zero_zero @ real ) ) )
=> ( Y6 = X5 ) ) ) ).
% cos_is_zero
thf(fact_2734_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_2735_cos__zero__iff,axiom,
! [X3: real] :
( ( ( cos @ real @ X3 )
= ( zero_zero @ real ) )
= ( ? [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X3
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X3
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% cos_zero_iff
thf(fact_2736_inverse__of__nat__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_2737_cos__expansion__lemma,axiom,
! [X3: real,M2: nat] :
( ( cos @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M2 ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( uminus_uminus @ real @ ( sin @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M2 ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_expansion_lemma
thf(fact_2738_tan__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( tan @ A @ X3 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_2739_exp__divide__power__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( power_power @ A @ ( exp @ A @ ( divide_divide @ A @ X3 @ ( semiring_1_of_nat @ A @ N ) ) ) @ N )
= ( exp @ A @ X3 ) ) ) ) ).
% exp_divide_power_eq
thf(fact_2740_le__log__of__power,axiom,
! [B2: real,N: nat,M2: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ B2 @ N ) @ M2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ M2 ) ) ) ) ).
% le_log_of_power
thf(fact_2741_log__base__pow,axiom,
! [A3: real,N: nat,X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( log @ ( power_power @ real @ A3 @ N ) @ X3 )
= ( divide_divide @ real @ ( log @ A3 @ X3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log_base_pow
thf(fact_2742_ln__realpow,axiom,
! [X3: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ln_ln @ real @ ( power_power @ real @ X3 @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( ln_ln @ real @ X3 ) ) ) ) ).
% ln_realpow
thf(fact_2743_log__nat__power,axiom,
! [X3: real,B2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( log @ B2 @ ( power_power @ real @ X3 @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ X3 ) ) ) ) ).
% log_nat_power
thf(fact_2744_cos__tan,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( cos @ real @ X3 )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_tan
thf(fact_2745_tan__45,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( one_one @ real ) ) ).
% tan_45
thf(fact_2746_cos__one__2pi__int,axiom,
! [X3: real] :
( ( ( cos @ real @ X3 )
= ( one_one @ real ) )
= ( ? [X4: int] :
( X3
= ( times_times @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ X4 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ).
% cos_one_2pi_int
thf(fact_2747_tan__60,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) ).
% tan_60
thf(fact_2748_cos__45,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% cos_45
thf(fact_2749_log2__of__power__eq,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( semiring_1_of_nat @ real @ N )
= ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ).
% log2_of_power_eq
thf(fact_2750_log__of__power__less,axiom,
! [M2: nat,B2: real,N: nat] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ M2 ) @ ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_2751_Bernoulli__inequality,axiom,
! [X3: real,N: nat] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X3 ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_2752_cos__plus__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z2: A] :
( ( plus_plus @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z2 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_2753_cos__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z2: A] :
( ( times_times @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( cos @ A @ ( minus_minus @ A @ W @ Z2 ) ) @ ( cos @ A @ ( plus_plus @ A @ W @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_cos
thf(fact_2754_cos__squared__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cos_squared_eq
thf(fact_2755_sin__squared__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_squared_eq
thf(fact_2756_lemma__tan__total,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ? [X5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X5 )
& ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ord_less @ real @ Y @ ( tan @ real @ X5 ) ) ) ) ).
% lemma_tan_total
thf(fact_2757_tan__gt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( tan @ real @ X3 ) ) ) ) ).
% tan_gt_zero
thf(fact_2758_lemma__tan__total1,axiom,
! [Y: real] :
? [X5: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X5 )
& ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X5 )
= Y ) ) ).
% lemma_tan_total1
thf(fact_2759_tan__mono__lt__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( tan @ real @ X3 ) @ ( tan @ real @ Y ) )
= ( ord_less @ real @ X3 @ Y ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_2760_tan__monotone_H,axiom,
! [Y: real,X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y @ X3 )
= ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X3 ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_2761_tan__monotone,axiom,
! [Y: real,X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X3 ) ) ) ) ) ).
% tan_monotone
thf(fact_2762_tan__total,axiom,
! [Y: real] :
? [X5: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X5 )
& ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X5 )
= Y )
& ! [Y6: real] :
( ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y6 )
& ( ord_less @ real @ Y6 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ Y6 )
= Y ) )
=> ( Y6 = X5 ) ) ) ).
% tan_total
thf(fact_2763_tan__minus__45,axiom,
( ( tan @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).
% tan_minus_45
thf(fact_2764_cos__double__less__one,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) ) @ ( one_one @ real ) ) ) ) ).
% cos_double_less_one
thf(fact_2765_tan__inverse,axiom,
! [Y: real] :
( ( divide_divide @ real @ ( one_one @ real ) @ ( tan @ real @ Y ) )
= ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y ) ) ) ).
% tan_inverse
thf(fact_2766_cos__gt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X3 ) ) ) ) ).
% cos_gt_zero
thf(fact_2767_cos__60,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% cos_60
thf(fact_2768_cos__30,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% cos_30
thf(fact_2769_log__of__power__le,axiom,
! [M2: nat,B2: real,N: nat] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ M2 ) @ ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less_eq @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_2770_cos__double__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ W ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ ( cos @ A @ W ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) ) ) ) ).
% cos_double_cos
thf(fact_2771_tan__cot_H,axiom,
! [X3: real] :
( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 ) )
= ( cot @ real @ X3 ) ) ).
% tan_cot'
thf(fact_2772_cos__treble__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ X3 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ ( cos @ A @ X3 ) ) ) ) ) ).
% cos_treble_cos
thf(fact_2773_cos__diff__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z2: A] :
( ( minus_minus @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z2 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ Z2 @ W ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_2774_sin__diff__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z2: A] :
( ( minus_minus @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z2 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_2775_sin__plus__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z2: A] :
( ( plus_plus @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z2 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_2776_cos__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z2: A] :
( ( times_times @ A @ ( cos @ A @ W ) @ ( sin @ A @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( sin @ A @ ( plus_plus @ A @ W @ Z2 ) ) @ ( sin @ A @ ( minus_minus @ A @ W @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_sin
thf(fact_2777_sin__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z2: A] :
( ( times_times @ A @ ( sin @ A @ W ) @ ( cos @ A @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( sin @ A @ ( plus_plus @ A @ W @ Z2 ) ) @ ( sin @ A @ ( minus_minus @ A @ W @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_cos
thf(fact_2778_sin__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z2: A] :
( ( times_times @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( cos @ A @ ( minus_minus @ A @ W @ Z2 ) ) @ ( cos @ A @ ( plus_plus @ A @ W @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_sin
thf(fact_2779_cos__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) )
= ( minus_minus @ A @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cos_double
thf(fact_2780_tan__pos__pi2__le,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tan @ real @ X3 ) ) ) ) ).
% tan_pos_pi2_le
thf(fact_2781_tan__total__pos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ? [X5: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X5 )
& ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X5 )
= Y ) ) ) ).
% tan_total_pos
thf(fact_2782_tan__less__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( tan @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% tan_less_zero
thf(fact_2783_tan__mono__le__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( tan @ real @ X3 ) @ ( tan @ real @ Y ) )
= ( ord_less_eq @ real @ X3 @ Y ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_2784_tan__mono__le,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( tan @ real @ X3 ) @ ( tan @ real @ Y ) ) ) ) ) ).
% tan_mono_le
thf(fact_2785_tan__bound__pi2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( abs_abs @ real @ ( tan @ real @ X3 ) ) @ ( one_one @ real ) ) ) ).
% tan_bound_pi2
thf(fact_2786_tan__30,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) ) ).
% tan_30
thf(fact_2787_cos__gt__zero__pi,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X3 ) ) ) ) ).
% cos_gt_zero_pi
thf(fact_2788_cos__ge__zero,axiom,
! [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 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cos @ real @ X3 ) ) ) ) ).
% cos_ge_zero
thf(fact_2789_arctan__unique,axiom,
! [X3: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( tan @ real @ X3 )
= Y )
=> ( ( arctan @ Y )
= X3 ) ) ) ) ).
% arctan_unique
thf(fact_2790_arctan__tan,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arctan @ ( tan @ real @ X3 ) )
= X3 ) ) ) ).
% arctan_tan
thf(fact_2791_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_2792_less__log2__of__power,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M2 )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ).
% less_log2_of_power
thf(fact_2793_le__log2__of__power,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M2 )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ).
% le_log2_of_power
thf(fact_2794_cos__double__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ W ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ ( sin @ A @ W ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_double_sin
thf(fact_2795_tan__total__pi4,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ? [Z3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) @ Z3 )
& ( ord_less @ real @ Z3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
& ( ( tan @ real @ Z3 )
= X3 ) ) ) ).
% tan_total_pi4
thf(fact_2796_cos__zero__iff__int,axiom,
! [X3: real] :
( ( ( cos @ real @ X3 )
= ( zero_zero @ real ) )
= ( ? [I4: int] :
( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I4 )
& ( X3
= ( times_times @ real @ ( ring_1_of_int @ real @ I4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_iff_int
thf(fact_2797_cos__arctan,axiom,
! [X3: real] :
( ( cos @ real @ ( arctan @ X3 ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_arctan
thf(fact_2798_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_2799_log2__of__power__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log2_of_power_less
thf(fact_2800_Bernoulli__inequality__even,axiom,
! [N: nat,X3: 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 ) @ X3 ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) @ N ) ) ) ).
% Bernoulli_inequality_even
thf(fact_2801_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_2802_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N: nat,X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ X3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X3 @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ X3 ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_2803_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X3: real,N: nat] :
( ( ord_less_eq @ real @ X3 @ ( 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 @ X3 @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ ( uminus_uminus @ real @ X3 ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_2804_sincos__total__pi,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X3 @ ( 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 )
& ( X3
= ( cos @ real @ T6 ) )
& ( Y
= ( sin @ real @ T6 ) ) ) ) ) ).
% sincos_total_pi
thf(fact_2805_sin__cos__sqrt,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) )
=> ( ( sin @ real @ X3 )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( cos @ real @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_cos_sqrt
thf(fact_2806_cos__arcsin,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X3 ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_arcsin
thf(fact_2807_log2__of__power__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less_eq @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log2_of_power_le
thf(fact_2808_sin__zero__lemma,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( sin @ real @ X3 )
= ( zero_zero @ real ) )
=> ? [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X3
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_2809_sin__zero__iff,axiom,
! [X3: real] :
( ( ( sin @ real @ X3 )
= ( zero_zero @ real ) )
= ( ? [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X3
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X3
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% sin_zero_iff
thf(fact_2810_sin__arccos,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X3 ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_arccos
thf(fact_2811_of__nat__code__if,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N3: nat] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [M5: nat,Q4: nat] :
( if @ A
@ ( Q4
= ( zero_zero @ nat ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M5 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M5 ) ) @ ( one_one @ A ) ) )
@ ( divmod_nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_2812_linear__plus__1__le__power,axiom,
! [X3: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X3 ) @ ( one_one @ real ) ) @ ( power_power @ real @ ( plus_plus @ real @ X3 @ ( one_one @ real ) ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_2813_monoseq__arctan__series,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( topological_monoseq @ real
@ ^ [N3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_2814_lemma__termdiff3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [H: A,Z2: A,K5: real,N: nat] :
( ( H
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K5 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z2 @ H ) ) @ K5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H ) @ N ) @ ( power_power @ A @ Z2 @ N ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) @ ( times_times @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( power_power @ real @ K5 @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_2815_ln__series,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ( ln_ln @ real @ X3 )
= ( suminf @ real
@ ^ [N3: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X3 @ ( one_one @ real ) ) @ ( suc @ N3 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_2816_arctan__series,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( arctan @ X3 )
= ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_2817_int__eq__iff__numeral,axiom,
! [M2: nat,V2: num] :
( ( ( semiring_1_of_nat @ int @ M2 )
= ( numeral_numeral @ int @ V2 ) )
= ( M2
= ( numeral_numeral @ nat @ V2 ) ) ) ).
% int_eq_iff_numeral
thf(fact_2818_negative__zless,axiom,
! [N: nat,M2: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) @ ( semiring_1_of_nat @ int @ M2 ) ) ).
% negative_zless
thf(fact_2819_powser__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F3: nat > A] :
( ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) ) )
= ( F3 @ ( zero_zero @ nat ) ) ) ) ).
% powser_zero
thf(fact_2820_int__ops_I3_J,axiom,
! [N: num] :
( ( semiring_1_of_nat @ int @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ int @ N ) ) ).
% int_ops(3)
thf(fact_2821_int__of__nat__induct,axiom,
! [P: int > $o,Z2: int] :
( ! [N2: nat] : ( P @ ( semiring_1_of_nat @ int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) )
=> ( P @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_2822_int__cases,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_2823_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% zle_int
thf(fact_2824_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A8: nat,B8: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A8 ) @ ( semiring_1_of_nat @ int @ B8 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_2825_zadd__int__left,axiom,
! [M2: nat,N: nat,Z2: int] :
( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M2 @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_2826_int__plus,axiom,
! [N: nat,M2: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N @ M2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M2 ) ) ) ).
% int_plus
thf(fact_2827_int__ops_I5_J,axiom,
! [A3: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A3 @ B2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_2828_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A8: nat,B8: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A8 ) @ ( semiring_1_of_nat @ int @ B8 ) ) ) ) ).
% nat_leq_as_int
thf(fact_2829_int__Suc,axiom,
! [N: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ N ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ).
% int_Suc
thf(fact_2830_int__ops_I4_J,axiom,
! [A3: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ A3 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( one_one @ int ) ) ) ).
% int_ops(4)
thf(fact_2831_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W3: int,Z4: int] :
? [N3: nat] :
( Z4
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_2832_lemma__NBseq__def2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X6: A > B] :
( ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X6 @ N3 ) ) @ K6 ) ) )
= ( ? [N6: nat] :
! [N3: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X6 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) ) ) ).
% lemma_NBseq_def2
thf(fact_2833_lemma__NBseq__def,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X6: A > B] :
( ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X6 @ N3 ) ) @ K6 ) ) )
= ( ? [N6: nat] :
! [N3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X6 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) ) ) ).
% lemma_NBseq_def
thf(fact_2834_monoseq__realpow,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( topological_monoseq @ real @ ( power_power @ real @ X3 ) ) ) ) ).
% monoseq_realpow
thf(fact_2835_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_2836_negD,axiom,
! [X3: int] :
( ( ord_less @ int @ X3 @ ( zero_zero @ int ) )
=> ? [N2: nat] :
( X3
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_2837_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_2838_zdiff__int__split,axiom,
! [P: int > $o,X3: nat,Y: nat] :
( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X3 @ Y ) ) )
= ( ( ( ord_less_eq @ nat @ Y @ X3 )
=> ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X3 ) @ ( semiring_1_of_nat @ int @ Y ) ) ) )
& ( ( ord_less @ nat @ X3 @ Y )
=> ( P @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_2839_exp__bound__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z2: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ Z2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% exp_bound_half
thf(fact_2840_exp__bound__lemma,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z2: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ Z2 ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( real_V7770717601297561774m_norm @ A @ Z2 ) ) ) ) ) ) ).
% exp_bound_lemma
thf(fact_2841_pi__series,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( suminf @ real
@ ^ [K3: nat] : ( divide_divide @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pi_series
thf(fact_2842_norm__divide__numeral,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: A,W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ W ) ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( numeral_numeral @ real @ W ) ) ) ) ).
% norm_divide_numeral
thf(fact_2843_norm__mult__numeral1,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [W: num,A3: A] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A3 ) )
= ( times_times @ real @ ( numeral_numeral @ real @ W ) @ ( real_V7770717601297561774m_norm @ A @ A3 ) ) ) ) ).
% norm_mult_numeral1
thf(fact_2844_norm__mult__numeral2,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: A,W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) )
= ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( numeral_numeral @ real @ W ) ) ) ) ).
% norm_mult_numeral2
thf(fact_2845_norm__neg__numeral,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( numeral_numeral @ real @ W ) ) ) ).
% norm_neg_numeral
thf(fact_2846_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_2847_norm__numeral,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( numeral_numeral @ A @ W ) )
= ( numeral_numeral @ real @ W ) ) ) ).
% norm_numeral
thf(fact_2848_norm__power,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X3: A,N: nat] :
( ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X3 @ N ) )
= ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ N ) ) ) ).
% norm_power
thf(fact_2849_norm__uminus__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,Y: A] :
( ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) )
= ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) ) ) ).
% norm_uminus_minus
thf(fact_2850_power__eq__imp__eq__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W: A,N: nat,Z2: A] :
( ( ( power_power @ A @ W @ N )
= ( power_power @ A @ Z2 @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( real_V7770717601297561774m_norm @ A @ W )
= ( real_V7770717601297561774m_norm @ A @ Z2 ) ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_2851_norm__triangle__lt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,Y: A,E3: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ E3 ) ) ) ).
% norm_triangle_lt
thf(fact_2852_norm__add__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,R2: real,Y: A,S: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( plus_plus @ real @ R2 @ S ) ) ) ) ) ).
% norm_add_less
thf(fact_2853_norm__power__ineq,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X3: A,N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X3 @ N ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ N ) ) ) ).
% norm_power_ineq
thf(fact_2854_norm__triangle__mono,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,R2: real,B2: A,S: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ R2 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ S )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( plus_plus @ real @ R2 @ S ) ) ) ) ) ).
% norm_triangle_mono
thf(fact_2855_norm__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) ) ) ).
% norm_triangle_ineq
thf(fact_2856_norm__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,Y: A,E3: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ E3 ) ) ) ).
% norm_triangle_le
thf(fact_2857_norm__add__leD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B2: A,C3: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A3 @ B2 ) ) @ C3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ C3 ) ) ) ) ).
% norm_add_leD
thf(fact_2858_norm__diff__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ).
% norm_diff_ineq
thf(fact_2859_power__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W: A,N: nat] :
( ( ( power_power @ A @ W @ N )
= ( one_one @ A ) )
=> ( ( ( real_V7770717601297561774m_norm @ A @ W )
= ( one_one @ real ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% power_eq_1_iff
thf(fact_2860_norm__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ C3 @ D3 ) ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A3 @ C3 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_2861_square__norm__one,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X3: A] :
( ( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ X3 )
= ( one_one @ real ) ) ) ) ).
% square_norm_one
thf(fact_2862_norm__power__diff,axiom,
! [A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z2: A,W: A,M2: nat] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ W ) @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( power_power @ A @ Z2 @ M2 ) @ ( power_power @ A @ W @ M2 ) ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M2 ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Z2 @ W ) ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_2863_ceiling__log__nat__eq__powr__iff,axiom,
! [B2: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K2 ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) )
= ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N ) @ K2 )
& ( ord_less_eq @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_2864_summable__arctan__series,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_2865_ceiling__log__nat__eq__if,axiom,
! [B2: nat,N: nat,K2: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N ) @ K2 )
=> ( ( ord_less_eq @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K2 ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_2866_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_2867_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M2: nat,X3: A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ( X3
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M2 ) ) ) )
& ( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X3 @ M2 ) @ ( power_power @ A @ X3 @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_2868_summable__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,K2: nat] :
( ( summable @ A
@ ^ [N3: nat] : ( F3 @ ( plus_plus @ nat @ N3 @ K2 ) ) )
= ( summable @ A @ F3 ) ) ) ).
% summable_iff_shift
thf(fact_2869_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A] :
( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty
thf(fact_2870_ceiling__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ int @ V2 ) ) ) ).
% ceiling_numeral
thf(fact_2871_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A6: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A6 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( abs_abs @ B @ ( F3 @ I4 ) )
@ A6 ) ) ) ).
% sum_abs
thf(fact_2872_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,X3: B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ~ ( member @ B @ X3 @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( insert2 @ B @ X3 @ A6 ) )
= ( plus_plus @ A @ ( G3 @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 ) ) ) ) ) ) ).
% sum.insert
thf(fact_2873_ceiling__add__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z2: int] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z2 ) ) ) ).
% ceiling_add_of_int
thf(fact_2874_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A6: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( abs_abs @ B @ ( F3 @ I4 ) )
@ A6 ) ) ) ).
% sum_abs_ge_zero
thf(fact_2875_ceiling__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) ) ) ).
% ceiling_le_zero
thf(fact_2876_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% ceiling_le_numeral
thf(fact_2877_ceiling__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) ) ) ).
% ceiling_less_one
thf(fact_2878_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V2 ) @ X3 ) ) ) ).
% numeral_less_ceiling
thf(fact_2879_ceiling__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X3 @ ( one_one @ A ) ) ) ) ).
% ceiling_le_one
thf(fact_2880_ceiling__add__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_add_numeral
thf(fact_2881_ceiling__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_neg_numeral
thf(fact_2882_ceiling__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) ) ) ) ).
% ceiling_add_one
thf(fact_2883_ceiling__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_diff_numeral
thf(fact_2884_ceiling__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: num,N: nat] :
( ( archimedean_ceiling @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ).
% ceiling_numeral_power
thf(fact_2885_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_2886_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M2: nat,G3: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_2887_ceiling__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_zero
thf(fact_2888_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A6: set @ nat,C3: nat > A] :
( ( ( ( finite_finite2 @ nat @ A6 )
& ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A6 )
= ( C3 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A6 )
& ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_2889_ceiling__divide__eq__div__numeral,axiom,
! [A3: num,B2: num] :
( ( archimedean_ceiling @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A3 ) @ ( numeral_numeral @ real @ B2 ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ).
% ceiling_divide_eq_div_numeral
thf(fact_2890_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_2891_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% numeral_le_ceiling
thf(fact_2892_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_2893_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X3 ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_2894_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A6: set @ nat,C3: nat > A,D3: nat > A] :
( ( ( ( finite_finite2 @ nat @ A6 )
& ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D3 @ I4 ) )
@ A6 )
= ( divide_divide @ A @ ( C3 @ ( zero_zero @ nat ) ) @ ( D3 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A6 )
& ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D3 @ I4 ) )
@ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_2895_ceiling__minus__divide__eq__div__numeral,axiom,
! [A3: num,B2: num] :
( ( archimedean_ceiling @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A3 ) @ ( numeral_numeral @ real @ B2 ) ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ A3 ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ).
% ceiling_minus_divide_eq_div_numeral
thf(fact_2896_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_2897_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_2898_summable__comparison__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G3: nat > real] :
( ? [N7: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N7 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) @ ( G3 @ N2 ) ) )
=> ( ( summable @ real @ G3 )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_comparison_test
thf(fact_2899_summable__comparison__test_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G3: nat > real,N5: nat,F3: nat > A] :
( ( summable @ real @ G3 )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) @ ( G3 @ N2 ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_comparison_test'
thf(fact_2900_summable__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,G3: nat > A] :
( ( summable @ A @ F3 )
=> ( ( summable @ A @ G3 )
=> ( summable @ A
@ ^ [N3: nat] : ( plus_plus @ A @ ( F3 @ N3 ) @ ( G3 @ N3 ) ) ) ) ) ) ).
% summable_add
thf(fact_2901_summable__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A
@ ^ [N3: nat] : ( F3 @ ( suc @ N3 ) ) )
= ( summable @ A @ F3 ) ) ) ).
% summable_Suc_iff
thf(fact_2902_summable__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,K2: nat] :
( ( summable @ A @ F3 )
=> ( summable @ A
@ ^ [N3: nat] : ( F3 @ ( plus_plus @ nat @ N3 @ K2 ) ) ) ) ) ).
% summable_ignore_initial_segment
thf(fact_2903_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K5: set @ B,F3: B > A,G3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ K5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G3 @ I3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ K5 ) ) ) ) ).
% sum_mono
thf(fact_2904_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A,H: B > A,A6: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( G3 @ X4 ) @ ( H @ X4 ) )
@ A6 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H @ A6 ) ) ) ) ).
% sum.distrib
thf(fact_2905_sum__le__suminf,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,I5: set @ nat] :
( ( summable @ A @ F3 )
=> ( ( finite_finite2 @ nat @ I5 )
=> ( ! [N2: nat] :
( ( member @ nat @ N2 @ ( uminus_uminus @ ( set @ nat ) @ I5 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ I5 ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ).
% sum_le_suminf
thf(fact_2906_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A6: set @ B,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_2907_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A6: set @ B,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) ) ) ) ).
% sum_nonneg
thf(fact_2908_powser__insidea,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,X3: A,Z2: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
=> ( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F3 @ N3 ) @ ( power_power @ A @ Z2 @ N3 ) ) ) ) ) ) ) ).
% powser_insidea
thf(fact_2909_sum__mono__inv,axiom,
! [A: $tType,I6: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [F3: I6 > A,I5: set @ I6,G3: I6 > A,I: I6] :
( ( ( groups7311177749621191930dd_sum @ I6 @ A @ F3 @ I5 )
= ( groups7311177749621191930dd_sum @ I6 @ A @ G3 @ I5 ) )
=> ( ! [I3: I6] :
( ( member @ I6 @ I3 @ I5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G3 @ I3 ) ) )
=> ( ( member @ I6 @ I @ I5 )
=> ( ( finite_finite2 @ I6 @ I5 )
=> ( ( F3 @ I )
= ( G3 @ I ) ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_2910_suminf__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,G3: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F3 @ N2 ) @ ( G3 @ N2 ) )
=> ( ( summable @ A @ F3 )
=> ( ( summable @ A @ G3 )
=> ( ord_less_eq @ A @ ( suminf @ A @ F3 ) @ ( suminf @ A @ G3 ) ) ) ) ) ) ).
% suminf_le
thf(fact_2911_sum__cong__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ nat,F3: nat > A,G3: nat > A] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ A6 )
=> ( ! [X5: nat] :
( ( member @ nat @ ( suc @ X5 ) @ A6 )
=> ( ( F3 @ ( suc @ X5 ) )
= ( G3 @ ( suc @ X5 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ A6 )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ A6 ) ) ) ) ) ).
% sum_cong_Suc
thf(fact_2912_ceiling__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ).
% ceiling_mono
thf(fact_2913_le__of__int__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ).
% le_of_int_ceiling
thf(fact_2914_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_2915_suminf__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,G3: nat > A] :
( ( summable @ A @ F3 )
=> ( ( summable @ A @ G3 )
=> ( ( plus_plus @ A @ ( suminf @ A @ F3 ) @ ( suminf @ A @ G3 ) )
= ( suminf @ A
@ ^ [N3: nat] : ( plus_plus @ A @ ( F3 @ N3 ) @ ( G3 @ N3 ) ) ) ) ) ) ) ).
% suminf_add
thf(fact_2916_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_2917_summable__partial__sum__bound,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,E3: real] :
( ( summable @ A @ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ~ ! [N8: nat] :
~ ! [M3: nat] :
( ( ord_less_eq @ nat @ N8 @ M3 )
=> ! [N9: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ M3 @ N9 ) ) ) @ E3 ) ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_2918_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( plus_plus @ nat @ I4 @ K2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_2919_suminf__nonneg,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_nonneg
thf(fact_2920_suminf__eq__zero__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ( ( suminf @ A @ F3 )
= ( zero_zero @ A ) )
= ( ! [N3: nat] :
( ( F3 @ N3 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_2921_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X5 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 )
= ( zero_zero @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ( F3 @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_2922_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,T2: set @ C,G3: C > A,I: C > B,F3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G3 @ X5 ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S )
=> ? [Xa: C] :
( ( member @ C @ Xa @ T2 )
& ( ( I @ Xa )
= X5 )
& ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G3 @ Xa ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S ) @ ( groups7311177749621191930dd_sum @ C @ A @ G3 @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_2923_sum__strict__mono__ex1,axiom,
! [A: $tType,I6: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A6: set @ I6,F3: I6 > A,G3: I6 > A] :
( ( finite_finite2 @ I6 @ A6 )
=> ( ! [X5: I6] :
( ( member @ I6 @ X5 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) )
=> ( ? [X: I6] :
( ( member @ I6 @ X @ A6 )
& ( ord_less @ A @ ( F3 @ X ) @ ( G3 @ X ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I6 @ A @ F3 @ A6 ) @ ( groups7311177749621191930dd_sum @ I6 @ A @ G3 @ A6 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_2924_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R: A > A > $o,S3: set @ B,H: B > A,G3: B > A] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X16: A,Y15: A,X22: A,Y23: A] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus @ A @ X16 @ Y15 ) @ ( plus_plus @ A @ X22 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( R @ ( H @ X5 ) @ ( G3 @ X5 ) ) )
=> ( R @ ( groups7311177749621191930dd_sum @ B @ A @ H @ S3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S3 ) ) ) ) ) ) ) ).
% sum.related
thf(fact_2925_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A6: set @ B,F3: B > A,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( ord_less @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_2926_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,X3: B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( ( member @ B @ X3 @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( insert2 @ B @ X3 @ A6 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 ) ) )
& ( ~ ( member @ B @ X3 @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( insert2 @ B @ X3 @ A6 ) )
= ( plus_plus @ A @ ( G3 @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_2927_ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,A3: int] :
( ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ A3 ) )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ A3 ) ) ) ).
% ceiling_le
thf(fact_2928_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z2: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z2 )
= ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% ceiling_le_iff
thf(fact_2929_summable__zero__power_H,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) ) ) ) ).
% summable_zero_power'
thf(fact_2930_summable__0__powser,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) ) ) ) ).
% summable_0_powser
thf(fact_2931_ceiling__add__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( archimedean_ceiling @ A @ Y ) ) ) ) ).
% ceiling_add_le
thf(fact_2932_powser__split__head_I3_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z2: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ N3 ) @ ( power_power @ A @ Z2 @ N3 ) ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ ( suc @ N3 ) ) @ ( power_power @ A @ Z2 @ N3 ) ) ) ) ) ).
% powser_split_head(3)
thf(fact_2933_summable__powser__split__head,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,Z2: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ ( suc @ N3 ) ) @ ( power_power @ A @ Z2 @ N3 ) ) )
= ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ N3 ) @ ( power_power @ A @ Z2 @ N3 ) ) ) ) ) ).
% summable_powser_split_head
thf(fact_2934_summable__powser__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,M2: nat,Z2: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ ( plus_plus @ nat @ N3 @ M2 ) ) @ ( power_power @ A @ Z2 @ N3 ) ) )
= ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ N3 ) @ ( power_power @ A @ Z2 @ N3 ) ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_2935_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F3: B > A,B5: A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S )
= B5 )
=> ( ( member @ B @ I @ S )
=> ( ord_less_eq @ A @ ( F3 @ I ) @ B5 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_2936_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F3: B > A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S )
=> ( ( F3 @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_2937_summable__norm__comparison__test,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G3: nat > real] :
( ? [N7: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N7 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) @ ( G3 @ N2 ) ) )
=> ( ( summable @ real @ G3 )
=> ( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_2938_summable__rabs__comparison__test,axiom,
! [F3: nat > real,G3: nat > real] :
( ? [N7: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N7 @ N2 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( F3 @ N2 ) ) @ ( G3 @ N2 ) ) )
=> ( ( summable @ real @ G3 )
=> ( summable @ real
@ ^ [N3: nat] : ( abs_abs @ real @ ( F3 @ N3 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_2939_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,M2: nat,I5: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X3 @ ( plus_plus @ nat @ M2 @ I4 ) )
@ I5 )
= ( times_times @ A @ ( power_power @ A @ X3 @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ I5 ) ) ) ) ).
% sum_power_add
thf(fact_2940_sum_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_2941_suminf__pos2,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,I: nat] :
( ( summable @ A @ F3 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ).
% suminf_pos2
thf(fact_2942_suminf__pos__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) )
= ( ? [I4: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I4 ) ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_2943_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 ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ I5 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_2944_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 ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ I5 ) ) ) ) ) ) ).
% sum_pos
thf(fact_2945_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S3: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X5 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G3 @ X5 )
= ( H @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ S3 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_2946_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S3: set @ B,H: B > A,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( H @ X5 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G3 @ X5 )
= ( H @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ T4 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_2947_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S3: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_2948_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S3: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ T4 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_2949_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C4: set @ B,A6: set @ B,B5: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C4 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C4 @ A6 ) )
=> ( ( G3 @ A5 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C4 @ B5 ) )
=> ( ( H @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ C4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ C4 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ B5 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_2950_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C4: set @ B,A6: set @ B,B5: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C4 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C4 @ A6 ) )
=> ( ( G3 @ A5 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C4 @ B5 ) )
=> ( ( H @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ B5 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ C4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ C4 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_2951_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B5: set @ B,A6: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ A6 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A6 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B5 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_2952_sum__diff,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A6: set @ B,B5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A6 @ B5 ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B5 ) ) ) ) ) ) ).
% sum_diff
thf(fact_2953_sum_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,B5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A6 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A6 @ B5 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B5 ) ) ) ) ) ) ).
% sum.union_inter
thf(fact_2954_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,G3: B > A,B5: set @ B] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A6 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A6 @ B5 ) ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_2955_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_2956_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_2957_sum__shift__lb__Suc0__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,K2: nat] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_2958_sum_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_2959_sum_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( G3 @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_2960_sum_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ ( suc @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_2961_powser__inside,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,X3: A,Z2: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ N3 ) @ ( power_power @ A @ Z2 @ N3 ) ) ) ) ) ) ).
% powser_inside
thf(fact_2962_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_2963_complete__algebra__summable__geometric,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ X3 ) ) ) ) ).
% complete_algebra_summable_geometric
thf(fact_2964_suminf__split__head,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( F3 @ ( suc @ N3 ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F3 ) @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% suminf_split_head
thf(fact_2965_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,P: B > $o,H: B > A,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( H @ X4 ) @ ( G3 @ X4 ) )
@ A6 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A6 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_2966_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G3 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ M2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_2967_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I4 ) ) @ ( F3 @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F3 @ ( suc @ N ) ) @ ( F3 @ M2 ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_2968_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B5: set @ B,A6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ B5 )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ B5 @ A6 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ B4 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B5 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_2969_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,B5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ A6 @ B5 ) )
=> ( ( G3 @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A6 @ B5 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B5 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_2970_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,G3: B > A,X3: B] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( insert2 @ B @ X3 @ A6 ) )
= ( plus_plus @ A @ ( G3 @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert2 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_2971_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,X3: B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( member @ B @ X3 @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 )
= ( plus_plus @ A @ ( G3 @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert2 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_2972_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A6: set @ B,A3: B,F3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( ( member @ B @ A3 @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) @ ( F3 @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_2973_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A6: set @ B,B5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A6 @ B5 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A6 @ B5 ) ) ) ) ) ) ) ).
% sum_Un
thf(fact_2974_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,B5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ( inf_inf @ ( set @ B ) @ A6 @ B5 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A6 @ B5 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B5 ) ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_2975_ceiling__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) @ ( one_one @ A ) ) @ X3 )
& ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ) ).
% ceiling_correct
thf(fact_2976_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X3: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) )
=> ( ( archimedean_ceiling @ A @ X3 )
= Z2 ) ) ) ) ).
% ceiling_unique
thf(fact_2977_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,A3: int] :
( ( ( archimedean_ceiling @ A @ X3 )
= A3 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) @ X3 )
& ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ A3 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_2978_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_2979_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [A6: set @ A,B5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ B5 @ A6 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ) ) ) ) ).
% sum_Un2
thf(fact_2980_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,B5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A6 @ B5 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A6 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ B5 @ A6 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A6 @ B5 ) ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_2981_mult__ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A3 @ B2 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A3 ) @ ( archimedean_ceiling @ A @ B2 ) ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_2982_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z2: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z2 )
= ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_2983_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A,P2: nat] :
( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P2 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_2984_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A6: set @ B,F3: B > A,B2: A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) @ B2 )
= ( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A8: B] : ( divide_divide @ A @ ( F3 @ A8 ) @ B2 )
@ ( inf_inf @ ( set @ B ) @ A6
@ ( collect @ B
@ ^ [A8: B] : ( dvd_dvd @ A @ B2 @ ( F3 @ A8 ) ) ) ) )
@ ( divide_divide @ A
@ ( groups7311177749621191930dd_sum @ B @ A @ F3
@ ( inf_inf @ ( set @ B ) @ A6
@ ( collect @ B
@ ^ [A8: B] :
~ ( dvd_dvd @ A @ B2 @ ( F3 @ A8 ) ) ) ) )
@ B2 ) ) ) ) ) ).
% sum_div_partition
thf(fact_2985_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A3: B,B2: B > A,C3: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C3 @ K3 ) )
@ S3 )
= ( plus_plus @ A @ ( B2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C3 @ K3 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_2986_set__encode__def,axiom,
( nat_set_encode
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% set_encode_def
thf(fact_2987_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B5: set @ A,A6: set @ A,B2: A,F3: A > B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B5 @ A6 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F3 @ B2 ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X5 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A6 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ B5 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_2988_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A6: set @ C,F3: C > B] :
( ( member @ C @ I @ A6 )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ ( minus_minus @ ( set @ C ) @ A6 @ ( insert2 @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X5 ) ) )
=> ( ( finite_finite2 @ C @ A6 )
=> ( ord_less_eq @ B @ ( F3 @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F3 @ A6 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_2989_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less_eq @ A @ P2 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P2 @ Q3 ) ) ) @ Q3 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_2990_powser__split__head_I1_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z2: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ N3 ) @ ( power_power @ A @ Z2 @ N3 ) ) )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ N3 ) @ ( power_power @ A @ Z2 @ N3 ) ) )
= ( plus_plus @ A @ ( F3 @ ( zero_zero @ nat ) )
@ ( times_times @ A
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ ( suc @ N3 ) ) @ ( power_power @ A @ Z2 @ N3 ) ) )
@ Z2 ) ) ) ) ) ).
% powser_split_head(1)
thf(fact_2991_powser__split__head_I2_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z2: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ N3 ) @ ( power_power @ A @ Z2 @ N3 ) ) )
=> ( ( times_times @ A
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ ( suc @ N3 ) ) @ ( power_power @ A @ Z2 @ N3 ) ) )
@ Z2 )
= ( minus_minus @ A
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ N3 ) @ ( power_power @ A @ Z2 @ N3 ) ) )
@ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% powser_split_head(2)
thf(fact_2992_suminf__exist__split,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,F3: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( summable @ A @ F3 )
=> ? [N8: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ N8 @ N9 )
=> ( ord_less @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( suminf @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N9 ) ) ) )
@ R2 ) ) ) ) ) ).
% suminf_exist_split
thf(fact_2993_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,X3: A > B,A3: A > B,B2: B,Delta: B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X3 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X3 @ I5 )
= ( one_one @ B ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I3 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( times_times @ B @ ( A3 @ I4 ) @ ( X3 @ I4 ) )
@ I5 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_2994_summable__power__series,axiom,
! [F3: nat > real,Z2: real] :
( ! [I3: nat] : ( ord_less_eq @ real @ ( F3 @ I3 ) @ ( one_one @ real ) )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ I3 ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Z2 )
=> ( ( ord_less @ real @ Z2 @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [I4: nat] : ( times_times @ real @ ( F3 @ I4 ) @ ( power_power @ real @ Z2 @ I4 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_2995_Abel__lemma,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,R0: real,A3: nat > A,M7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( ord_less @ real @ R2 @ R0 )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A3 @ N2 ) ) @ ( power_power @ real @ R0 @ N2 ) ) @ M7 )
=> ( summable @ real
@ ^ [N3: nat] : ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A3 @ N3 ) ) @ ( power_power @ real @ R2 @ N3 ) ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_2996_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F3: nat > A] :
( ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F3 @ M2 ) @ ( F3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_2997_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) )
= ( minus_minus @ A @ ( F3 @ N ) @ ( F3 @ M2 ) ) ) ) ) ).
% sum_telescope''
thf(fact_2998_summable__ratio__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [C3: real,N5: nat,F3: nat > A] :
( ( ord_less @ real @ C3 @ ( one_one @ real ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ ( suc @ N2 ) ) ) @ ( times_times @ real @ C3 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_ratio_test
thf(fact_2999_ceiling__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: int,X3: A] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ N ) @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ N ) @ ( one_one @ A ) ) )
=> ( ( archimedean_ceiling @ A @ X3 )
= ( plus_plus @ int @ N @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_eq
thf(fact_3000_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q4: nat] : ( ord_less @ nat @ Q4 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_3001_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M2: nat,N: nat,X3: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X3 @ M2 ) @ ( power_power @ A @ X3 @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_3002_sum_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.in_pairs
thf(fact_3003_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q4: nat] : ( ord_less @ nat @ Q4 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_3004_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% gauss_sum_nat
thf(fact_3005_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_3006_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,D3: 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 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) ) ) ).
% double_arith_series
thf(fact_3007_arith__series__nat,axiom,
! [A3: nat,D3: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ A3 @ ( times_times @ nat @ I4 @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ nat @ N @ D3 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_3008_Sum__Icc__nat,axiom,
! [M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M2 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Icc_nat
thf(fact_3009_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,D3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_3010_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_3011_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_3012_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,M2: nat,N: nat] :
( ( ( X3
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ N ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
& ( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X3 @ M2 ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_3013_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_3014_lemma__termdiff2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [H: A,Z2: A,N: nat] :
( ( H
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H ) @ N ) @ ( power_power @ A @ Z2 @ N ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
= ( times_times @ A @ H
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [Q4: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H ) @ Q4 ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q4 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ P5 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_3015_ceiling__log__eq__powr__iff,axiom,
! [X3: real,B2: real,K2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archimedean_ceiling @ real @ ( log @ B2 @ X3 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ K2 ) @ ( one_one @ int ) ) )
= ( ( ord_less @ real @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ K2 ) ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_3016_geometric__deriv__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Z2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( one_one @ real ) )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) @ ( power_power @ A @ Z2 @ N3 ) )
@ ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( minus_minus @ A @ ( one_one @ A ) @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_3017_monoseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X8: nat > A] :
( ! [M5: nat,N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
| ! [M5: nat,N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) ) ) ) ) ) ).
% monoseq_def
thf(fact_3018_monoI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X6: nat > A] :
( ! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( X6 @ M ) ) )
=> ( topological_monoseq @ A @ X6 ) ) ) ).
% monoI2
thf(fact_3019_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X3 ) @ ( set_ord_lessThan @ A @ Y ) )
= ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% lessThan_subset_iff
thf(fact_3020_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_3021_sum_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( G3 @ N ) ) ) ) ).
% sum.lessThan_Suc
thf(fact_3022_single__Diff__lessThan,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K2: A] :
( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ K2 @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K2 ) )
= ( insert2 @ A @ K2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% single_Diff_lessThan
thf(fact_3023_numeral__powr__numeral__real,axiom,
! [M2: num,N: num] :
( ( powr @ real @ ( numeral_numeral @ real @ M2 ) @ ( numeral_numeral @ real @ N ) )
= ( power_power @ real @ ( numeral_numeral @ real @ M2 ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% numeral_powr_numeral_real
thf(fact_3024_powser__sums__zero__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: nat > A,X3: A] :
( ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A3 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) )
@ X3 )
= ( ( A3 @ ( zero_zero @ nat ) )
= X3 ) ) ) ).
% powser_sums_zero_iff
thf(fact_3025_powr__numeral,axiom,
! [X3: real,N: num] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( powr @ real @ X3 @ ( numeral_numeral @ real @ N ) )
= ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% powr_numeral
thf(fact_3026_square__powr__half,axiom,
! [X3: real] :
( ( powr @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( abs_abs @ real @ X3 ) ) ).
% square_powr_half
thf(fact_3027_sums__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,G3: nat > A,S: A,T2: A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F3 @ N2 ) @ ( G3 @ N2 ) )
=> ( ( sums @ A @ F3 @ S )
=> ( ( sums @ A @ G3 @ T2 )
=> ( ord_less_eq @ A @ S @ T2 ) ) ) ) ) ).
% sums_le
thf(fact_3028_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P: A > nat,N: A] :
( ! [X5: A] : ( ord_less_eq @ nat @ ( Q @ X5 ) @ ( P @ X5 ) )
=> ( ( 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
@ ^ [X4: A] : ( minus_minus @ nat @ ( P @ X4 ) @ ( Q @ X4 ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_3029_lessThan__non__empty,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X3: A] :
( ( set_ord_lessThan @ A @ X3 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% lessThan_non_empty
thf(fact_3030_sums__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,A3: A,G3: nat > A,B2: A] :
( ( sums @ A @ F3 @ A3 )
=> ( ( sums @ A @ G3 @ B2 )
=> ( sums @ A
@ ^ [N3: nat] : ( plus_plus @ A @ ( F3 @ N3 ) @ ( G3 @ N3 ) )
@ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% sums_add
thf(fact_3031_sums__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,N: nat,S: A] :
( ( sums @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N ) )
@ S )
= ( sums @ A @ F3 @ ( plus_plus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_iff_shift
thf(fact_3032_sums__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S: A,N: nat] :
( ( sums @ A @ F3 @ S )
=> ( sums @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_3033_sums__iff__shift_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,N: nat,S: A] :
( ( sums @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) ) )
= ( sums @ A @ F3 @ S ) ) ) ).
% sums_iff_shift'
thf(fact_3034_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_3035_lessThan__Suc,axiom,
! [K2: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K2 ) )
= ( insert2 @ nat @ K2 @ ( set_ord_lessThan @ nat @ K2 ) ) ) ).
% lessThan_Suc
thf(fact_3036_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_3037_sum__subtractf__nat,axiom,
! [A: $tType,A6: set @ A,G3: A > nat,F3: A > nat] :
( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ nat @ ( G3 @ X5 ) @ ( F3 @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( minus_minus @ nat @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ A6 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A6 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G3 @ A6 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_3038_sum__eq__Suc0__iff,axiom,
! [A: $tType,A6: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A6 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ( F3 @ X4 )
= ( suc @ ( zero_zero @ nat ) ) )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( ( X4 != Y3 )
=> ( ( F3 @ Y3 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_3039_sum__SucD,axiom,
! [A: $tType,F3: A > nat,A6: set @ A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A6 )
= ( suc @ N ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A6 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X5 ) ) ) ) ).
% sum_SucD
thf(fact_3040_sums__Suc__imp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S: A] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N3: nat] : ( F3 @ ( suc @ N3 ) )
@ S )
=> ( sums @ A @ F3 @ S ) ) ) ) ).
% sums_Suc_imp
thf(fact_3041_sums__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S: A] :
( ( sums @ A
@ ^ [N3: nat] : ( F3 @ ( suc @ N3 ) )
@ S )
= ( sums @ A @ F3 @ ( plus_plus @ A @ S @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc_iff
thf(fact_3042_sums__Suc,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,L: A] :
( ( sums @ A
@ ^ [N3: nat] : ( F3 @ ( suc @ N3 ) )
@ L )
=> ( sums @ A @ F3 @ ( plus_plus @ A @ L @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc
thf(fact_3043_sums__zero__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [N: nat,F3: nat > A,S: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( F3 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N ) )
@ S )
= ( sums @ A @ F3 @ S ) ) ) ) ).
% sums_zero_iff_shift
thf(fact_3044_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_3045_powr__add,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X3: A,A3: A,B2: A] :
( ( powr @ A @ X3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( powr @ A @ X3 @ A3 ) @ ( powr @ A @ X3 @ B2 ) ) ) ) ).
% powr_add
thf(fact_3046_lessThan__nat__numeral,axiom,
! [K2: num] :
( ( set_ord_lessThan @ nat @ ( numeral_numeral @ nat @ K2 ) )
= ( insert2 @ nat @ ( pred_numeral @ K2 ) @ ( set_ord_lessThan @ nat @ ( pred_numeral @ K2 ) ) ) ) ).
% lessThan_nat_numeral
thf(fact_3047_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_3048_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X4: complex] : X4
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= ( one_one @ complex ) ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_roots_unity
thf(fact_3049_sum__nth__roots,axiom,
! [N: nat,C3: complex] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X4: complex] : X4
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= C3 ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_nth_roots
thf(fact_3050_powser__sums__if,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [M2: nat,Z2: A] :
( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( if @ A @ ( N3 = M2 ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z2 @ N3 ) )
@ ( power_power @ A @ Z2 @ M2 ) ) ) ).
% powser_sums_if
thf(fact_3051_powser__sums__zero,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: nat > A] :
( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A3 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) )
@ ( A3 @ ( zero_zero @ nat ) ) ) ) ).
% powser_sums_zero
thf(fact_3052_sum__diff__nat,axiom,
! [A: $tType,B5: set @ A,A6: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A6 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ B5 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_3053_sum__diff1__nat,axiom,
! [A: $tType,A3: A,A6: set @ A,F3: A > nat] :
( ( ( member @ A @ A3 @ A6 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A6 ) @ ( F3 @ A3 ) ) ) )
& ( ~ ( member @ A @ A3 @ A6 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A6 ) ) ) ) ).
% sum_diff1_nat
thf(fact_3054_powr__realpow,axiom,
! [X3: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( powr @ real @ X3 @ ( semiring_1_of_nat @ real @ N ) )
= ( power_power @ real @ X3 @ N ) ) ) ).
% powr_realpow
thf(fact_3055_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,K2: A] :
( ( ( ord_less @ A @ X3 @ K2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K2 ) @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X3 @ K2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K2 ) @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_3056_sums__If__finite__set_H,axiom,
! [A: $tType] :
( ( ( topolo1287966508704411220up_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G3: nat > A,S3: A,A6: set @ nat,S5: A,F3: nat > A] :
( ( sums @ A @ G3 @ S3 )
=> ( ( finite_finite2 @ nat @ A6 )
=> ( ( S5
= ( plus_plus @ A @ S3
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F3 @ N3 ) @ ( G3 @ N3 ) )
@ A6 ) ) )
=> ( sums @ A
@ ^ [N3: nat] : ( if @ A @ ( member @ nat @ N3 @ A6 ) @ ( F3 @ N3 ) @ ( G3 @ N3 ) )
@ S5 ) ) ) ) ) ).
% sums_If_finite_set'
thf(fact_3057_suminf__le__const,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,X3: A] :
( ( summable @ A @ F3 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N2 ) ) @ X3 )
=> ( ord_less_eq @ A @ ( suminf @ A @ F3 ) @ X3 ) ) ) ) ).
% suminf_le_const
thf(fact_3058_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_3059_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ M2 ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_3060_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N3 ) ) @ ( F3 @ N3 ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( minus_minus @ A @ ( F3 @ M2 ) @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_3061_summableI__nonneg__bounded,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,X3: A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N2 ) ) @ X3 )
=> ( summable @ A @ F3 ) ) ) ) ).
% summableI_nonneg_bounded
thf(fact_3062_sum_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( G3 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_3063_sum__Un__nat,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A6 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_3064_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X3 @ N ) @ ( one_one @ A ) )
= ( times_times @ A @ ( minus_minus @ A @ X3 @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_3065_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_3066_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,N: nat] :
( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X3 @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X3 @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_3067_suminf__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,K2: nat] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A @ F3 )
= ( plus_plus @ A
@ ( suminf @ A
@ ^ [N3: nat] : ( F3 @ ( plus_plus @ nat @ N3 @ K2 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ K2 ) ) ) ) ) ) ).
% suminf_split_initial_segment
thf(fact_3068_suminf__minus__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,K2: nat] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( F3 @ ( plus_plus @ nat @ N3 @ K2 ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ K2 ) ) ) ) ) ) ).
% suminf_minus_initial_segment
thf(fact_3069_sum__less__suminf,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,N: nat] :
( ( summable @ A @ F3 )
=> ( ! [M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ M ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% sum_less_suminf
thf(fact_3070_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( ( X3
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) )
& ( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_3071_lemma__termdiff1,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [Z2: A,H: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] : ( minus_minus @ A @ ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H ) @ ( minus_minus @ nat @ M2 @ P5 ) ) @ ( power_power @ A @ Z2 @ P5 ) ) @ ( power_power @ A @ Z2 @ M2 ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] : ( times_times @ A @ ( power_power @ A @ Z2 @ P5 ) @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H ) @ ( minus_minus @ nat @ M2 @ P5 ) ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ M2 @ P5 ) ) ) )
@ ( set_ord_lessThan @ nat @ M2 ) ) ) ) ).
% lemma_termdiff1
thf(fact_3072_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X3 @ N ) @ ( power_power @ A @ Y @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X3 @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) ) @ ( power_power @ A @ X3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_3073_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X3 @ ( suc @ N ) ) @ ( power_power @ A @ Y @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X3 @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] : ( times_times @ A @ ( power_power @ A @ X3 @ P5 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ P5 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_3074_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_3075_power__half__series,axiom,
( sums @ real
@ ^ [N3: nat] : ( power_power @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( suc @ N3 ) )
@ ( one_one @ real ) ) ).
% power_half_series
thf(fact_3076_real__sum__nat__ivl__bounded2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,F3: nat > A,K5: A,K2: nat] :
( ! [P6: nat] :
( ( ord_less @ nat @ P6 @ N )
=> ( ord_less_eq @ A @ ( F3 @ P6 ) @ K5 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K5 )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ K2 ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ K5 ) ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_3077_sum__less__suminf2,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,N: nat,I: nat] :
( ( summable @ A @ F3 )
=> ( ! [M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ M ) ) )
=> ( ( ord_less_eq @ nat @ N @ 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_3078_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X3 @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_3079_sums__if_H,axiom,
! [G3: nat > real,X3: real] :
( ( sums @ real @ G3 @ X3 )
=> ( sums @ real
@ ^ [N3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( zero_zero @ real ) @ ( G3 @ ( divide_divide @ nat @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ X3 ) ) ).
% sums_if'
thf(fact_3080_sums__if,axiom,
! [G3: nat > real,X3: real,F3: nat > real,Y: real] :
( ( sums @ real @ G3 @ X3 )
=> ( ( sums @ real @ F3 @ Y )
=> ( sums @ real
@ ^ [N3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( F3 @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( G3 @ ( divide_divide @ nat @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ real @ X3 @ Y ) ) ) ) ).
% sums_if
thf(fact_3081_powr__half__sqrt,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( powr @ real @ X3 @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( sqrt @ X3 ) ) ) ).
% powr_half_sqrt
thf(fact_3082_powr__neg__numeral,axiom,
! [X3: real,N: num] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( powr @ real @ X3 @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ N ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_3083_sum__split__even__odd,axiom,
! [F3: nat > real,G3: nat > real,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( F3 @ I4 ) @ ( G3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( F3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( G3 @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( one_one @ nat ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum_split_even_odd
thf(fact_3084_Sum__Icc__int,axiom,
! [M2: int,N: int] :
( ( ord_less_eq @ int @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X4: int] : X4
@ ( set_or1337092689740270186AtMost @ int @ M2 @ N ) )
= ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N @ ( plus_plus @ int @ N @ ( one_one @ int ) ) ) @ ( times_times @ int @ M2 @ ( minus_minus @ int @ M2 @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_3085_sum__pos__lt__pair,axiom,
! [F3: nat > real,K2: nat] :
( ( summable @ real @ F3 )
=> ( ! [D2: nat] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( F3 @ ( plus_plus @ nat @ K2 @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D2 ) ) ) @ ( F3 @ ( plus_plus @ nat @ K2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D2 ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ord_less @ real @ ( groups7311177749621191930dd_sum @ nat @ real @ F3 @ ( set_ord_lessThan @ nat @ K2 ) ) @ ( suminf @ real @ F3 ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_3086_mono__SucI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X6: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
=> ( topological_monoseq @ A @ X6 ) ) ) ).
% mono_SucI1
thf(fact_3087_mono__SucI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X6: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) )
=> ( topological_monoseq @ A @ X6 ) ) ) ).
% mono_SucI2
thf(fact_3088_monoseq__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X8: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
| ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) ) ) ) ) ) ).
% monoseq_Suc
thf(fact_3089_monoI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X6: nat > A] :
( ! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ M ) @ ( X6 @ N2 ) ) )
=> ( topological_monoseq @ A @ X6 ) ) ) ).
% monoI1
thf(fact_3090_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( F3 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_3091_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ ( zero_zero @ real ) @ M5 ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( one_one @ real ) ) ).
% sumr_cos_zero_one
thf(fact_3092_diffs__equiv,axiom,
! [A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( ring_1 @ A ) )
=> ! [C3: nat > A,X3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( C3 @ N3 ) ) @ ( power_power @ A @ X3 @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) ) ) ) ) ).
% diffs_equiv
thf(fact_3093_finite__nat__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S6: set @ nat] :
? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S6 @ ( set_ord_lessThan @ nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_3094_finite__nat__bounded,axiom,
! [S3: set @ nat] :
( ( finite_finite2 @ nat @ S3 )
=> ? [K: nat] : ( ord_less_eq @ ( set @ nat ) @ S3 @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% finite_nat_bounded
thf(fact_3095_diffs__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( diffs @ A )
= ( ^ [C6: nat > A,N3: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) @ ( C6 @ ( suc @ N3 ) ) ) ) ) ) ).
% diffs_def
thf(fact_3096_termdiff__converges__all,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,X3: A] :
( ! [X5: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C3 @ N3 ) @ ( power_power @ A @ X5 @ N3 ) ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) ) ) ) ).
% termdiff_converges_all
thf(fact_3097_finite__transitivity__chain,axiom,
! [A: $tType,A6: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [X5: A] :
~ ( R @ X5 @ X5 )
=> ( ! [X5: A,Y4: A,Z3: A] :
( ( R @ X5 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X5 @ Z3 ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A6 )
& ( R @ X5 @ Y6 ) ) )
=> ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_3098_termdiff__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,K5: real,C3: nat > A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K5 )
=> ( ! [X5: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K5 )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C3 @ N3 ) @ ( power_power @ A @ X5 @ N3 ) ) ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) ) ) ) ) ).
% termdiff_converges
thf(fact_3099_infinite__nat__iff__unbounded__le,axiom,
! [S3: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S3 ) )
= ( ! [M5: nat] :
? [N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
& ( member @ nat @ N3 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_3100_Maclaurin__cos__expansion2,axiom,
! [X3: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less @ real @ T6 @ X3 )
& ( ( cos @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ X3 @ M5 ) )
@ ( 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 @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_3101_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ? [T6: real] :
( ( ord_less @ real @ X3 @ T6 )
& ( ord_less @ real @ T6 @ ( zero_zero @ real ) )
& ( ( cos @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ X3 @ M5 ) )
@ ( 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 @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_3102_Maclaurin__cos__expansion,axiom,
! [X3: real,N: nat] :
? [T6: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T6 ) @ ( abs_abs @ real @ X3 ) )
& ( ( cos @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ X3 @ M5 ) )
@ ( 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 @ X3 @ N ) ) ) ) ) ).
% Maclaurin_cos_expansion
thf(fact_3103_sin__paired,axiom,
! [X3: real] :
( sums @ real
@ ^ [N3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) )
@ ( sin @ real @ X3 ) ) ).
% sin_paired
thf(fact_3104_arcosh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A )
= ( ^ [X4: A] : ( ln_ln @ A @ ( plus_plus @ A @ X4 @ ( powr @ A @ ( minus_minus @ A @ ( power_power @ A @ X4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% arcosh_def
thf(fact_3105_of__real__numeral,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [W: num] :
( ( real_Vector_of_real @ A @ ( numeral_numeral @ real @ W ) )
= ( numeral_numeral @ A @ W ) ) ) ).
% of_real_numeral
thf(fact_3106_of__real__power,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X3: real,N: nat] :
( ( real_Vector_of_real @ A @ ( power_power @ real @ X3 @ N ) )
= ( power_power @ A @ ( real_Vector_of_real @ A @ X3 ) @ N ) ) ) ).
% of_real_power
thf(fact_3107_of__real__add,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X3: real,Y: real] :
( ( real_Vector_of_real @ A @ ( plus_plus @ real @ X3 @ Y ) )
= ( plus_plus @ A @ ( real_Vector_of_real @ A @ X3 ) @ ( real_Vector_of_real @ A @ Y ) ) ) ) ).
% of_real_add
thf(fact_3108_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_3109_fact__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ ( suc @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_Suc
thf(fact_3110_fact__2,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% fact_2
thf(fact_3111_of__real__neg__numeral,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [W: num] :
( ( real_Vector_of_real @ A @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ W ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ).
% of_real_neg_numeral
thf(fact_3112_norm__of__real__add1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X3: real] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X3 ) @ ( one_one @ A ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X3 @ ( one_one @ real ) ) ) ) ) ).
% norm_of_real_add1
thf(fact_3113_norm__of__real__addn,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X3: real,B2: num] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X3 ) @ ( numeral_numeral @ A @ B2 ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X3 @ ( numeral_numeral @ real @ B2 ) ) ) ) ) ).
% norm_of_real_addn
thf(fact_3114_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_3115_sin__of__real__pi__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V7773925162809079976_field @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_of_real_pi_half
thf(fact_3116_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_3117_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_3118_fact__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ M2 ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_mono
thf(fact_3119_fact__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( dvd_dvd @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M2 ) ) ) ) ).
% fact_dvd
thf(fact_3120_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: nat,N: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K2 ) @ ( semiring_char_0_fact @ A @ N ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K2 @ N ) ) ) ) ).
% fact_fact_dvd_fact
thf(fact_3121_fact__mod,axiom,
! [A: $tType] :
( ( ( linordered_semidom @ A )
& ( semidom_modulo @ A ) )
=> ! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( modulo_modulo @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M2 ) )
= ( zero_zero @ A ) ) ) ) ).
% fact_mod
thf(fact_3122_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_3123_norm__less__p1,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X3: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) @ ( one_one @ A ) ) ) ) ) ).
% norm_less_p1
thf(fact_3124_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K2 ) ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% choose_dvd
thf(fact_3125_fact__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K2: num] :
( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ K2 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ K2 ) @ ( semiring_char_0_fact @ A @ ( pred_numeral @ K2 ) ) ) ) ) ).
% fact_numeral
thf(fact_3126_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_3127_cos__sin__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cos @ A )
= ( ^ [X4: A] : ( sin @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X4 ) ) ) ) ) ).
% cos_sin_eq
thf(fact_3128_sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sin @ A )
= ( ^ [X4: A] : ( cos @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X4 ) ) ) ) ) ).
% sin_cos_eq
thf(fact_3129_Maclaurin__zero,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X3: real,N: nat,Diff: nat > A > real] :
( ( X3
= ( zero_zero @ real ) )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X3 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% Maclaurin_zero
thf(fact_3130_Maclaurin__lemma,axiom,
! [H: real,F3: real > real,J: nat > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H )
=> ? [B7: real] :
( ( F3 @ H )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( J @ M5 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ B7 @ ( divide_divide @ real @ ( power_power @ real @ H @ N ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_3131_minus__sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( uminus_uminus @ A @ ( sin @ A @ X3 ) )
= ( cos @ A @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% minus_sin_cos_eq
thf(fact_3132_Maclaurin__exp__le,axiom,
! [X3: real,N: nat] :
? [T6: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T6 ) @ ( abs_abs @ real @ X3 ) )
& ( ( exp @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( divide_divide @ real @ ( power_power @ real @ X3 @ M5 ) @ ( semiring_char_0_fact @ real @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X3 @ N ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_3133_cos__coeff__def,axiom,
( cos_coeff
= ( ^ [N3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( zero_zero @ real ) ) ) ) ).
% cos_coeff_def
thf(fact_3134_cos__paired,axiom,
! [X3: real] :
( sums @ real
@ ^ [N3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( semiring_char_0_fact @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) @ ( power_power @ real @ X3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( cos @ real @ X3 ) ) ).
% cos_paired
thf(fact_3135_arsinh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A )
= ( ^ [X4: A] : ( ln_ln @ A @ ( plus_plus @ A @ X4 @ ( powr @ A @ ( plus_plus @ A @ ( power_power @ A @ X4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% arsinh_def
thf(fact_3136_Maclaurin__exp__lt,axiom,
! [X3: real,N: nat] :
( ( X3
!= ( 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 @ X3 ) )
& ( ( exp @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( divide_divide @ real @ ( power_power @ real @ X3 @ M5 ) @ ( semiring_char_0_fact @ real @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_3137_Maclaurin__sin__expansion3,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ? [T6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less @ real @ T6 @ X3 )
& ( ( sin @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X3 @ M5 ) )
@ ( 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 @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_3138_Maclaurin__sin__expansion4,axiom,
! [X3: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ? [T6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less_eq @ real @ T6 @ X3 )
& ( ( sin @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X3 @ M5 ) )
@ ( 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 @ X3 @ N ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_3139_Maclaurin__sin__expansion2,axiom,
! [X3: real,N: nat] :
? [T6: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T6 ) @ ( abs_abs @ real @ X3 ) )
& ( ( sin @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X3 @ M5 ) )
@ ( 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 @ X3 @ N ) ) ) ) ) ).
% Maclaurin_sin_expansion2
thf(fact_3140_Maclaurin__sin__expansion,axiom,
! [X3: real,N: nat] :
? [T6: real] :
( ( sin @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X3 @ M5 ) )
@ ( 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 @ X3 @ N ) ) ) ) ).
% Maclaurin_sin_expansion
thf(fact_3141_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N3 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_3142_fact__mono__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( semiring_char_0_fact @ nat @ M2 ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ).
% fact_mono_nat
thf(fact_3143_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( semiring_char_0_fact @ nat @ N ) ) ).
% fact_ge_self
thf(fact_3144_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_3145_dvd__fact,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( dvd_dvd @ nat @ M2 @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% dvd_fact
thf(fact_3146_fact__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ N @ ( suc @ M2 ) )
=> ( ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N ) )
= ( times_times @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ M2 @ N ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_3147_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_3148_sin__coeff__Suc,axiom,
! [N: nat] :
( ( sin_coeff @ ( suc @ N ) )
= ( divide_divide @ real @ ( cos_coeff @ N ) @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) ) ).
% sin_coeff_Suc
thf(fact_3149_cos__coeff__Suc,axiom,
! [N: nat] :
( ( cos_coeff @ ( suc @ N ) )
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( sin_coeff @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) ) ).
% cos_coeff_Suc
thf(fact_3150_floor__log__nat__eq__powr__iff,axiom,
! [B2: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K2 ) ) )
= ( semiring_1_of_nat @ int @ N ) )
= ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N ) @ K2 )
& ( ord_less @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_3151_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% pochhammer_double
thf(fact_3152_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N3: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I4: A] : ( plus_plus @ A @ I4 @ ( one_one @ A ) )
@ N3
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_3153_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R2: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ R2 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M2 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R2 @ ( suc @ M2 ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_3154_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_3155_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_3156_gbinomial__0_I2_J,axiom,
! [B: $tType] :
( ( ( semiring_char_0 @ B )
& ( semidom_divide @ B ) )
=> ! [K2: nat] :
( ( gbinomial @ B @ ( zero_zero @ B ) @ ( suc @ K2 ) )
= ( zero_zero @ B ) ) ) ).
% gbinomial_0(2)
thf(fact_3157_floor__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archim6421214686448440834_floor @ A @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ int @ V2 ) ) ) ).
% floor_numeral
thf(fact_3158_binomial__0__Suc,axiom,
! [K2: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K2 ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_3159_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
= N ) ).
% binomial_1
thf(fact_3160_binomial__Suc__Suc,axiom,
! [N: nat,K2: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K2 ) )
= ( plus_plus @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_3161_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_3162_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% pochhammer_Suc0
thf(fact_3163_zero__less__binomial__iff,axiom,
! [N: nat,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K2 ) )
= ( ord_less_eq @ nat @ K2 @ N ) ) ).
% zero_less_binomial_iff
thf(fact_3164_zero__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ) ).
% zero_le_floor
thf(fact_3165_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V2 ) @ X3 ) ) ) ).
% numeral_le_floor
thf(fact_3166_zero__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ).
% zero_less_floor
thf(fact_3167_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% floor_less_numeral
thf(fact_3168_one__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ).
% one_le_floor
thf(fact_3169_floor__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% floor_neg_numeral
thf(fact_3170_floor__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% floor_diff_numeral
thf(fact_3171_floor__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: num,N: nat] :
( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ).
% floor_numeral_power
thf(fact_3172_floor__divide__eq__div__numeral,axiom,
! [A3: num,B2: num] :
( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A3 ) @ ( numeral_numeral @ real @ B2 ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ A3 ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_divide_eq_div_numeral
thf(fact_3173_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% numeral_less_floor
thf(fact_3174_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_3175_one__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) ) ) ).
% one_less_floor
thf(fact_3176_floor__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) )
= ( ord_less @ A @ X3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% floor_le_one
thf(fact_3177_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X3 ) ) ) ).
% neg_numeral_le_floor
thf(fact_3178_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_3179_floor__one__divide__eq__div__numeral,axiom,
! [B2: num] :
( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ B2 ) ) )
= ( divide_divide @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_one_divide_eq_div_numeral
thf(fact_3180_floor__minus__divide__eq__div__numeral,axiom,
! [A3: num,B2: num] :
( ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A3 ) @ ( numeral_numeral @ real @ B2 ) ) ) )
= ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_minus_divide_eq_div_numeral
thf(fact_3181_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% neg_numeral_less_floor
thf(fact_3182_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_3183_floor__minus__one__divide__eq__div__numeral,axiom,
! [B2: num] :
( ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ B2 ) ) ) )
= ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_minus_one_divide_eq_div_numeral
thf(fact_3184_floor__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ).
% floor_mono
thf(fact_3185_of__int__floor__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) @ X3 ) ) ).
% of_int_floor_le
thf(fact_3186_Suc__times__binomial__eq,axiom,
! [N: nat,K2: nat] :
( ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) )
= ( times_times @ nat @ ( binomial @ ( suc @ N ) @ ( suc @ K2 ) ) @ ( suc @ K2 ) ) ) ).
% Suc_times_binomial_eq
thf(fact_3187_Suc__times__binomial,axiom,
! [K2: nat,N: nat] :
( ( times_times @ nat @ ( suc @ K2 ) @ ( binomial @ ( suc @ N ) @ ( suc @ K2 ) ) )
= ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) ) ) ).
% Suc_times_binomial
thf(fact_3188_binomial__symmetric,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( binomial @ N @ K2 )
= ( binomial @ N @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ).
% binomial_symmetric
thf(fact_3189_choose__mult__lemma,axiom,
! [M2: nat,R2: nat,K2: nat] :
( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M2 @ R2 ) @ K2 ) @ ( plus_plus @ nat @ M2 @ K2 ) ) @ ( binomial @ ( plus_plus @ nat @ M2 @ K2 ) @ K2 ) )
= ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M2 @ R2 ) @ K2 ) @ K2 ) @ ( binomial @ ( plus_plus @ nat @ M2 @ R2 ) @ M2 ) ) ) ).
% choose_mult_lemma
thf(fact_3190_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_3191_pochhammer__eq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat,M2: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( comm_s3205402744901411588hammer @ A @ A3 @ M2 )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_3192_pochhammer__neq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ M2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
!= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_3193_zero__less__binomial,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K2 ) ) ) ).
% zero_less_binomial
thf(fact_3194_le__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X3: A] :
( ( ord_less_eq @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 ) ) ) ).
% le_floor_iff
thf(fact_3195_Suc__times__binomial__add,axiom,
! [A3: nat,B2: nat] :
( ( times_times @ nat @ ( suc @ A3 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B2 ) ) @ ( suc @ A3 ) ) )
= ( times_times @ nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B2 ) ) @ A3 ) ) ) ).
% Suc_times_binomial_add
thf(fact_3196_binomial__Suc__Suc__eq__times,axiom,
! [N: nat,K2: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K2 ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) ) @ ( suc @ K2 ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_3197_le__floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) ) ) ) ).
% le_floor_add
thf(fact_3198_int__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X3: A] :
( ( plus_plus @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 ) ) ) ) ).
% int_add_floor
thf(fact_3199_floor__add__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z2: int] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z2 )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ) ).
% floor_add_int
thf(fact_3200_choose__mult,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( times_times @ nat @ ( binomial @ N @ M2 ) @ ( binomial @ M2 @ K2 ) )
= ( times_times @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ ( minus_minus @ nat @ N @ K2 ) @ ( minus_minus @ nat @ M2 @ K2 ) ) ) ) ) ) ).
% choose_mult
thf(fact_3201_binomial__fact__lemma,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K2 ) ) ) @ ( binomial @ N @ K2 ) )
= ( semiring_char_0_fact @ nat @ N ) ) ) ).
% binomial_fact_lemma
thf(fact_3202_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A8: A,K3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A8 ) @ K3 ) ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_3203_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A8: A,K3: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A8 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( one_one @ A ) ) @ K3 ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_3204_floor__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,N: nat] :
( ( X3
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ X3 @ N ) )
= ( power_power @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ N ) ) ) ) ).
% floor_power
thf(fact_3205_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K2: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K2 ) )
= ( plus_plus @ A @ ( gbinomial @ A @ A3 @ K2 ) @ ( gbinomial @ A @ A3 @ ( suc @ K2 ) ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_3206_gbinomial__of__nat__symmetric,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K2 )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_3207_pochhammer__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X3 @ N ) ) ) ) ).
% pochhammer_nonneg
thf(fact_3208_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_3209_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_3210_one__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% one_add_floor
thf(fact_3211_binomial__absorption,axiom,
! [K2: nat,N: nat] :
( ( times_times @ nat @ ( suc @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K2 ) ) ) ).
% binomial_absorption
thf(fact_3212_binomial__altdef__nat,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( binomial @ N @ K2 )
= ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_3213_gbinomial__addition__formula,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K2: nat] :
( ( gbinomial @ A @ A3 @ ( suc @ K2 ) )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K2 ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% gbinomial_addition_formula
thf(fact_3214_gbinomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K2: nat,A3: A] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K2 ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K2 ) ) @ K2 ) @ ( gbinomial @ A @ A3 @ K2 ) ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_3215_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K2: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K2 ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K2 ) @ ( gbinomial @ A @ A3 @ K2 ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K2 ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_3216_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K2: nat] :
( ( times_times @ A @ A3 @ ( gbinomial @ A @ A3 @ K2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K2 ) @ ( gbinomial @ A @ A3 @ K2 ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K2 ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_3217_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ A3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% pochhammer_rec
thf(fact_3218_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A3 @ N ) @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_3219_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) )
= ( times_times @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) ) ) ) ).
% pochhammer_rec'
thf(fact_3220_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K2 )
!= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_3221_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N: nat,M2: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( plus_plus @ nat @ N @ M2 ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ M2 ) ) ) ) ).
% pochhammer_product'
thf(fact_3222_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_3223_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,A3: int] :
( ( ( archim6421214686448440834_floor @ A @ X3 )
= A3 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ X3 )
& ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_3224_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X3: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 )
=> ( ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X3 )
= Z2 ) ) ) ) ).
% floor_unique
thf(fact_3225_binomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_of_nat @ A @ K2 ) ) @ K2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_3226_le__mult__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A3 ) @ ( archim6421214686448440834_floor @ A @ B2 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ).
% le_mult_floor
thf(fact_3227_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X3: A] :
( ( ord_less @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% less_floor_iff
thf(fact_3228_binomial__mono,axiom,
! [K2: nat,K7: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ K7 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ K7 ) ) ) ) ).
% binomial_mono
thf(fact_3229_binomial__maximum_H,axiom,
! [N: nat,K2: nat] : ( ord_less_eq @ nat @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K2 ) @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ).
% binomial_maximum'
thf(fact_3230_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z2: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z2 )
= ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_3231_binomial__maximum,axiom,
! [N: nat,K2: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% binomial_maximum
thf(fact_3232_binomial__antimono,axiom,
! [K2: nat,K7: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ K7 )
=> ( ( ord_less_eq @ nat @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ K2 )
=> ( ( ord_less_eq @ nat @ K7 @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K7 ) @ ( binomial @ N @ K2 ) ) ) ) ) ).
% binomial_antimono
thf(fact_3233_binomial__le__pow2,axiom,
! [N: nat,K2: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% binomial_le_pow2
thf(fact_3234_floor__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) @ X3 )
& ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_correct
thf(fact_3235_choose__reduce__nat,axiom,
! [N: nat,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( binomial @ N @ K2 )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K2 ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_3236_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K2 ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A3 @ K2 ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_3237_gbinomial__absorption,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K2 ) ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% gbinomial_absorption
thf(fact_3238_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,M2: nat,A3: A] :
( ( ord_less_eq @ nat @ K2 @ M2 )
=> ( ( times_times @ A @ ( gbinomial @ A @ A3 @ M2 ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M2 ) @ K2 ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( minus_minus @ nat @ M2 @ K2 ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_3239_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M2: nat,N: nat,Z2: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ Z2 @ N )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ M2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_3240_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P2 @ Q3 ) ) ) @ Q3 ) @ P2 ) ) ) ).
% floor_divide_lower
thf(fact_3241_binomial__less__binomial__Suc,axiom,
! [K2: nat,N: nat] :
( ( ord_less @ nat @ K2 @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) ) ) ).
% binomial_less_binomial_Suc
thf(fact_3242_binomial__strict__mono,axiom,
! [K2: nat,K7: nat,N: nat] :
( ( ord_less @ nat @ K2 @ K7 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ K7 ) ) ) ) ).
% binomial_strict_mono
thf(fact_3243_binomial__strict__antimono,axiom,
! [K2: nat,K7: nat,N: nat] :
( ( ord_less @ nat @ K2 @ K7 )
=> ( ( ord_less_eq @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 ) )
=> ( ( ord_less_eq @ nat @ K7 @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K7 ) @ ( binomial @ N @ K2 ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_3244_central__binomial__odd,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( binomial @ N @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( binomial @ N @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% central_binomial_odd
thf(fact_3245_binomial__addition__formula,axiom,
! [N: nat,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( binomial @ N @ ( suc @ K2 ) )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( suc @ K2 ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K2 ) ) ) ) ).
% binomial_addition_formula
thf(fact_3246_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_3247_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ) ).
% fact_binomial
thf(fact_3248_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K2: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K2 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) ) @ ( gbinomial @ A @ A3 @ K2 ) ) ) ) ).
% gbinomial_factors
thf(fact_3249_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K2: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K2 ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K2 ) @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_3250_gbinomial__index__swap,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ K2 ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( one_one @ A ) ) @ N ) ) ) ) ).
% gbinomial_index_swap
thf(fact_3251_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A8: A,K3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A8 ) @ ( one_one @ A ) ) @ K3 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_3252_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R2: A,K2: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R2 ) @ K2 ) )
= ( times_times @ A @ R2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R2 ) @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_3253_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less @ A @ P2 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P2 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) ) ) ) ).
% floor_divide_upper
thf(fact_3254_pochhammer__same,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( comm_ring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% pochhammer_same
thf(fact_3255_choose__two,axiom,
! [N: nat] :
( ( binomial @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% choose_two
thf(fact_3256_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X4: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X4 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_3257_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K2: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K2 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% gbinomial_minus
thf(fact_3258_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( gbinomial @ A @ A3 @ K2 )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K2 ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_3259_pochhammer__minus_H,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K2: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( one_one @ A ) ) @ K2 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K2 ) ) ) ) ).
% pochhammer_minus'
thf(fact_3260_pochhammer__minus,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K2: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K2 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% pochhammer_minus
thf(fact_3261_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_3262_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_3263_fact__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_double
thf(fact_3264_floor__log__nat__eq__if,axiom,
! [B2: nat,N: nat,K2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N ) @ K2 )
=> ( ( ord_less @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K2 ) ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_3265_binomial__code,axiom,
( binomial
= ( ^ [N3: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N3 @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N3 @ ( minus_minus @ nat @ N3 @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N3 @ K3 ) @ ( one_one @ nat ) ) @ N3 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_3266_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_3267_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A8: A,N3: nat] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A8 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_3268_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A3 @ ( plus_plus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_3269_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_3270_atMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K2: A] :
( ( member @ A @ I @ ( set_ord_atMost @ A @ K2 ) )
= ( ord_less_eq @ A @ I @ K2 ) ) ) ).
% atMost_iff
thf(fact_3271_prod_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: B > A] :
( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty
thf(fact_3272_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X3 ) @ ( set_ord_atMost @ A @ Y ) )
= ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% atMost_subset_iff
thf(fact_3273_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,X3: B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ~ ( member @ B @ X3 @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert2 @ B @ X3 @ A6 ) )
= ( times_times @ A @ ( G3 @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 ) ) ) ) ) ) ).
% prod.insert
thf(fact_3274_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_3275_sum_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ).
% sum.atMost_Suc
thf(fact_3276_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( G3 @ N ) ) ) ) ).
% prod.lessThan_Suc
thf(fact_3277_prod_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ).
% prod.atMost_Suc
thf(fact_3278_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_3279_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G3: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_3280_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_3281_prod__power__distrib,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ B )
=> ! [F3: A > B,A6: set @ A,N: nat] :
( ( power_power @ B @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ A6 ) @ N )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N )
@ A6 ) ) ) ).
% prod_power_distrib
thf(fact_3282_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_3283_prod_Onested__swap_H,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_ord_lessThan @ nat @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( A3 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nested_swap'
thf(fact_3284_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X4: A] : ( ord_less_eq @ A @ X4 @ U2 ) ) ) ) ) ).
% atMost_def
thf(fact_3285_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A6: set @ B,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) ) ) ) ).
% prod_nonneg
thf(fact_3286_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A6: set @ B,F3: B > A,G3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) )
& ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G3 @ I3 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 ) ) ) ) ).
% prod_mono
thf(fact_3287_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A6: set @ B,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F3 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) ) ) ) ).
% prod_ge_1
thf(fact_3288_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_3289_lessThan__Suc__atMost,axiom,
! [K2: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K2 ) )
= ( set_ord_atMost @ nat @ K2 ) ) ).
% lessThan_Suc_atMost
thf(fact_3290_atMost__Suc,axiom,
! [K2: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K2 ) )
= ( insert2 @ nat @ ( suc @ K2 ) @ ( set_ord_atMost @ nat @ K2 ) ) ) ).
% atMost_Suc
thf(fact_3291_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_3292_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_3293_power__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C3: A,F3: B > nat,A6: set @ B] :
( ( power_power @ A @ C3 @ ( groups7311177749621191930dd_sum @ B @ nat @ F3 @ A6 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [A8: B] : ( power_power @ A @ C3 @ ( F3 @ A8 ) )
@ A6 ) ) ) ).
% power_sum
thf(fact_3294_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( plus_plus @ nat @ I4 @ K2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_3295_finite__nat__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S6: set @ nat] :
? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S6 @ ( set_ord_atMost @ nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_3296_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A6: set @ B,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X5 ) )
& ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_3297_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,X3: B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( ( member @ B @ X3 @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert2 @ B @ X3 @ A6 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 ) ) )
& ( ~ ( member @ B @ X3 @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert2 @ B @ X3 @ A6 ) )
= ( times_times @ A @ ( G3 @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_3298_prod__dvd__prod__subset2,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B5: set @ B,A6: set @ B,F3: B > A,G3: B > A] :
( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ B5 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ A6 )
=> ( dvd_dvd @ A @ ( F3 @ A5 ) @ ( G3 @ A5 ) ) )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B5 ) ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_3299_prod__dvd__prod__subset,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B5: set @ B,A6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ B5 )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_3300_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_3301_atMost__nat__numeral,axiom,
! [K2: num] :
( ( set_ord_atMost @ nat @ ( numeral_numeral @ nat @ K2 ) )
= ( insert2 @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( set_ord_atMost @ nat @ ( pred_numeral @ K2 ) ) ) ) ).
% atMost_nat_numeral
thf(fact_3302_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_3303_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% Iic_subset_Iio_iff
thf(fact_3304_prod_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat,M2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_3305_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P2: nat,K2: nat,G3: nat > A,H: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P2 )
=> ( ( ord_less_eq @ nat @ K2 @ P2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K2 ) @ ( G3 @ J3 ) @ ( if @ A @ ( J3 = K2 ) @ ( one_one @ A ) @ ( H @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K2 ) @ ( G3 @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_3306_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 ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ I5 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_3307_sum__choose__upper,axiom,
! [M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ K3 @ M2 )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ N ) @ ( suc @ M2 ) ) ) ).
% sum_choose_upper
thf(fact_3308_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 ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ I5 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_3309_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B5: set @ B,A6: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ A6 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A6 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B5 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_3310_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S3: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X5 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G3 @ X5 )
= ( H @ X5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ S3 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_3311_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S3: set @ B,H: B > A,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( H @ X5 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G3 @ X5 )
= ( H @ X5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ T4 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_3312_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S3: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_3313_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S3: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ T4 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_3314_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C4: set @ B,A6: set @ B,B5: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C4 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C4 @ A6 ) )
=> ( ( G3 @ A5 )
= ( one_one @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C4 @ B5 ) )
=> ( ( H @ B4 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ C4 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ C4 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ B5 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_3315_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C4: set @ B,A6: set @ B,B5: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C4 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C4 @ A6 ) )
=> ( ( G3 @ A5 )
= ( one_one @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C4 @ B5 ) )
=> ( ( H @ B4 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ B5 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ C4 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ C4 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_3316_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_3317_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_3318_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G3 @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_3319_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_3320_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G3 @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ M2 )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_3321_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_3322_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_3323_prod_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( G3 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_3324_polyfun__eq__coeffs,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat,D3: nat > A] :
( ( ! [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ X4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( D3 @ I4 ) @ ( power_power @ A @ X4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
= ( ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
=> ( ( C3 @ I4 )
= ( D3 @ I4 ) ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_3325_bounded__imp__summable,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linord2810124833399127020strict @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A3: nat > A,B5: A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A3 @ ( set_ord_atMost @ nat @ N2 ) ) @ B5 )
=> ( summable @ A @ A3 ) ) ) ) ).
% bounded_imp_summable
thf(fact_3326_sum_Onested__swap_H,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_ord_lessThan @ nat @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( A3 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nested_swap'
thf(fact_3327_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_3328_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_3329_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A6: set @ B,F3: B > A,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) )
& ( ord_less @ A @ ( F3 @ I3 ) @ ( G3 @ I3 ) ) ) )
=> ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_3330_even__prod__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_3331_sum__choose__lower,axiom,
! [R2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( plus_plus @ nat @ R2 @ K3 ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus @ nat @ R2 @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_3332_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,G3: B > A,X3: B] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert2 @ B @ X3 @ A6 ) )
= ( times_times @ A @ ( G3 @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert2 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_3333_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,X3: B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( member @ B @ X3 @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 )
= ( times_times @ A @ ( G3 @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert2 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_3334_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,B5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ( inf_inf @ ( set @ B ) @ A6 @ B5 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A6 @ B5 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B5 ) ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_3335_choose__rising__sum_I2_J,axiom,
! [N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ ( one_one @ nat ) ) @ M2 ) ) ).
% choose_rising_sum(2)
thf(fact_3336_choose__rising__sum_I1_J,axiom,
! [N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% choose_rising_sum(1)
thf(fact_3337_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A,P2: nat] :
( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P2 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_3338_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F4: nat > A > A,A8: nat,B8: nat,Acc2: A] : ( if @ A @ ( ord_less @ nat @ B8 @ A8 ) @ Acc2 @ ( set_fo6178422350223883121st_nat @ A @ F4 @ ( plus_plus @ nat @ A8 @ ( one_one @ nat ) ) @ B8 @ ( F4 @ A8 @ Acc2 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_3339_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X3: nat > A > A,Xa2: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X3 @ Xa2 @ Xb @ Xc )
= Y )
=> ( ( ( ord_less @ nat @ Xb @ Xa2 )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa2 )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X3 @ ( plus_plus @ nat @ Xa2 @ ( one_one @ nat ) ) @ Xb @ ( X3 @ Xa2 @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_3340_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A3: B,B2: B > A,C3: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C3 @ K3 ) )
@ S3 )
= ( times_times @ A @ ( B2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C3 @ K3 ) )
@ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_3341_zero__polynom__imp__zero__coeffs,axiom,
! [A: $tType] :
( ( ( ab_semigroup_mult @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [C3: nat > A,N: nat,K2: nat] :
( ! [W2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ W2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( C3 @ K2 )
= ( zero_zero @ A ) ) ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_3342_polyfun__eq__0,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat] :
( ( ! [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ X4 @ 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_3343_sum_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_3344_sum__up__index__split,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M2 @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_atMost @ nat @ M2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum_up_index_split
thf(fact_3345_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K3 ) ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% gbinomial_parallel_sum
thf(fact_3346_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_3347_fact__eq__fact__times,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( semiring_char_0_fact @ nat @ M2 )
= ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N )
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_3348_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B5: set @ A,A6: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ ( minus_minus @ ( set @ A ) @ B5 @ A6 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F3 @ B4 ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ A5 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ A6 ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ B5 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_3349_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A6: set @ B,F3: B > A,A3: B] :
( ( finite_finite2 @ B @ A6 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A3 @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) @ ( F3 @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_3350_sum__choose__diagonal,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( minus_minus @ nat @ N @ K3 ) @ ( minus_minus @ nat @ M2 @ K3 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( binomial @ ( suc @ N ) @ M2 ) ) ) ).
% sum_choose_diagonal
thf(fact_3351_vandermonde,axiom,
! [M2: nat,N: nat,R2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( binomial @ M2 @ K3 ) @ ( binomial @ N @ ( minus_minus @ nat @ R2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R2 ) )
= ( binomial @ ( plus_plus @ nat @ M2 @ N ) @ R2 ) ) ).
% vandermonde
thf(fact_3352_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N ) ) ) ) ) ).
% sum_gp_basic
thf(fact_3353_polyfun__roots__finite,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,K2: nat,N: nat] :
( ( ( C3 @ K2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_3354_polyfun__finite__roots,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ X4 @ 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_3355_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_3356_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 ) )
=> ~ ! [B4: nat > A] :
~ ! [Z5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z5 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ Z5 @ A3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( B4 @ I4 ) @ ( power_power @ A @ Z5 @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_3357_polyfun__linear__factor,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C3: nat > A,N: nat,A3: A] :
? [B4: nat > A] :
! [Z5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z5 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A
@ ( times_times @ A @ ( minus_minus @ A @ Z5 @ A3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( B4 @ I4 ) @ ( power_power @ A @ Z5 @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ A3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% polyfun_linear_factor
thf(fact_3358_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M2: nat,N: nat,X3: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( power_power @ A @ X3 @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_3359_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A8: A,N3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A8 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N3 @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N3 ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_3360_fact__div__fact,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M2 ) @ ( semiring_char_0_fact @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M2 ) ) ) ) ).
% fact_div_fact
thf(fact_3361_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_3362_choose__row__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ N ) @ ( set_ord_atMost @ nat @ N ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% choose_row_sum
thf(fact_3363_binomial,axiom,
! [A3: nat,B2: nat,N: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N @ K3 ) ) @ ( power_power @ nat @ A3 @ K3 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ).
% binomial
thf(fact_3364_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_3365_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_3366_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,A3: nat,B2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A8: nat] : ( plus_plus @ A @ ( F3 @ A8 ) )
@ A3
@ B2
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_3367_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.in_pairs_0
thf(fact_3368_polynomial__product,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [M2: nat,A3: nat > A,N: nat,B2: nat > A,X3: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ M2 @ I3 )
=> ( ( A3 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ A ) ) )
=> ( ( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ X3 @ I4 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( times_times @ A @ ( B2 @ J3 ) @ ( power_power @ A @ X3 @ 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 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ A @ X3 @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ) ) ).
% polynomial_product
thf(fact_3369_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_3370_polyfun__eq__const,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat,K2: A] :
( ( ! [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ X4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= K2 ) )
= ( ( ( C3 @ ( zero_zero @ nat ) )
= K2 )
& ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) )
=> ( ( C3 @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_const
thf(fact_3371_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ M2 ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_3372_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( power_power @ A @ A3 @ K3 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% binomial_ring
thf(fact_3373_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ A3 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_3374_polynomial__product__nat,axiom,
! [M2: nat,A3: nat > nat,N: nat,B2: nat > nat,X3: nat] :
( ! [I3: nat] :
( ( ord_less @ nat @ M2 @ I3 )
=> ( ( A3 @ I3 )
= ( zero_zero @ nat ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ nat ) ) )
=> ( ( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ ( A3 @ I4 ) @ ( power_power @ nat @ X3 @ I4 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( times_times @ nat @ ( B2 @ J3 ) @ ( power_power @ nat @ X3 @ 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 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ nat @ X3 @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_3375_choose__square__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( power_power @ nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ).
% choose_square_sum
thf(fact_3376_gbinomial__Suc,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A,K2: nat] :
( ( gbinomial @ A @ A3 @ ( suc @ K2 ) )
= ( 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 ) @ K2 ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K2 ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_3377_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P2: nat,K2: nat,G3: nat > A,H: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P2 )
=> ( ( ord_less_eq @ nat @ K2 @ P2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K2 ) @ ( G3 @ J3 ) @ ( if @ A @ ( J3 = K2 ) @ ( zero_zero @ A ) @ ( H @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K2 ) @ ( G3 @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_3378_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A3: A,X3: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A3 ) @ K3 ) @ ( power_power @ A @ X3 @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( minus_minus @ nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_3379_root__polyfun,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,Z2: A,A3: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( ( power_power @ A @ Z2 @ N )
= A3 )
= ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [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 @ Z2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% root_polyfun
thf(fact_3380_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( ( X3
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
& ( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ).
% sum_gp0
thf(fact_3381_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_3382_gbinomial__sum__nat__pow2,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M2 @ K3 ) ) @ K3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_3383_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A3: A,X3: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A3 ) @ K3 ) @ ( power_power @ A @ X3 @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A3 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X3 @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( minus_minus @ nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_3384_polyfun__diff__alt,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A3: nat > A,X3: 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 @ X3 @ 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 @ X3 @ 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 @ X3 @ J3 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ J3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_3385_binomial__r__part__sum,axiom,
! [M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M2 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ).
% binomial_r_part_sum
thf(fact_3386_choose__linear__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ I4 @ ( binomial @ N @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% choose_linear_sum
thf(fact_3387_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_3388_polyfun__extremal__lemma,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [E3: real,C3: nat > A,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M8: real] :
! [Z5: A] :
( ( ord_less_eq @ real @ M8 @ ( real_V7770717601297561774m_norm @ A @ Z5 ) )
=> ( ord_less_eq @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z5 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
@ ( times_times @ real @ E3 @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_3389_polyfun__diff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A3: nat > A,X3: 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 @ X3 @ 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 @ X3 @ 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 @ X3 @ J3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff
thf(fact_3390_fact__code,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 @ ( one_one @ nat ) ) ) ) ) ) ).
% fact_code
thf(fact_3391_gbinomial__r__part__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M2 ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ).
% gbinomial_r_part_sum
thf(fact_3392_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_3393_sin__x__sin__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( sums @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
@ ( 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 @ N3 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) ) @ ( power_power @ A @ X3 @ N3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P5 @ N3 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ Y ) ) ) ) ).
% sin_x_sin_y
thf(fact_3394_sums__cos__x__plus__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( sums @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: 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 @ N3 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ A @ X3 @ N3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P5 @ N3 ) ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( cos @ A @ ( plus_plus @ A @ X3 @ Y ) ) ) ) ).
% sums_cos_x_plus_y
thf(fact_3395_cos__x__cos__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( sums @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 )
& ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
@ ( 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 @ N3 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ A @ X3 @ N3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P5 @ N3 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) ) ) ).
% cos_x_cos_y
thf(fact_3396_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X4: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X4 ) ) @ ( archimedean_ceiling @ A @ X4 ) @ ( archim6421214686448440834_floor @ A @ X4 ) ) ) ) ) ).
% round_altdef
thf(fact_3397_Maclaurin__sin__bound,axiom,
! [X3: real,N: nat] :
( ord_less_eq @ real
@ ( abs_abs @ real
@ ( minus_minus @ real @ ( sin @ real @ X3 )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X3 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) )
@ ( times_times @ real @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( abs_abs @ real @ X3 ) @ N ) ) ) ).
% Maclaurin_sin_bound
thf(fact_3398_of__nat__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( ^ [N3: nat] : N3 ) ) ).
% of_nat_id
thf(fact_3399_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_3400_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_3401_scaleR__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: A,U: real,A3: A] :
( ( ( plus_plus @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ U @ A3 ) )
= ( plus_plus @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ U @ B2 ) ) )
= ( ( A3 = B2 )
| ( U
= ( one_one @ real ) ) ) ) ) ).
% scaleR_eq_iff
thf(fact_3402_scaleR__power,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X3: real,Y: A,N: nat] :
( ( power_power @ A @ ( real_V8093663219630862766scaleR @ A @ X3 @ Y ) @ N )
= ( real_V8093663219630862766scaleR @ A @ ( power_power @ real @ X3 @ N ) @ ( power_power @ A @ Y @ N ) ) ) ) ).
% scaleR_power
thf(fact_3403_inverse__le__iff__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% inverse_le_iff_le
thf(fact_3404_inverse__le__iff__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_3405_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num] :
( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_3406_scaleR__collapse,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,A3: A] :
( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U ) @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ U @ A3 ) )
= A3 ) ) ).
% scaleR_collapse
thf(fact_3407_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_3408_scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U: num,W: num,A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ U ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A3 ) )
= ( real_V8093663219630862766scaleR @ A @ ( times_times @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ W ) ) @ A3 ) ) ) ).
% scaleR_times
thf(fact_3409_inverse__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [V2: num,W: num,A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A3 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ W ) @ ( numeral_numeral @ real @ V2 ) ) @ A3 ) ) ) ).
% inverse_scaleR_times
thf(fact_3410_fraction__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U: num,V2: num,W: num,A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A3 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ W ) ) @ ( numeral_numeral @ real @ V2 ) ) @ A3 ) ) ) ).
% fraction_scaleR_times
thf(fact_3411_scaleR__half__double,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ A3 @ A3 ) )
= A3 ) ) ).
% scaleR_half_double
thf(fact_3412_power__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( inverse_inverse @ A @ A3 ) @ N )
= ( inverse_inverse @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_inverse
thf(fact_3413_scaleR__right__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X3: A,Y: A] :
( ( real_V8093663219630862766scaleR @ A @ A3 @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y ) ) ) ) ).
% scaleR_right_distrib
thf(fact_3414_real__vector__affinity__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M2: real,X3: A,C3: A,Y: A] :
( ( M2
!= ( zero_zero @ real ) )
=> ( ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M2 @ X3 ) @ C3 )
= Y )
= ( X3
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M2 ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M2 ) @ C3 ) ) ) ) ) ) ).
% real_vector_affinity_eq
thf(fact_3415_real__vector__eq__affinity,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M2: real,Y: A,X3: A,C3: A] :
( ( M2
!= ( zero_zero @ real ) )
=> ( ( Y
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M2 @ X3 ) @ C3 ) )
= ( ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M2 ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M2 ) @ C3 ) )
= X3 ) ) ) ) ).
% real_vector_eq_affinity
thf(fact_3416_neg__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% neg_le_divideR_eq
thf(fact_3417_neg__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) @ A3 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ B2 ) ) ) ) ).
% neg_divideR_le_eq
thf(fact_3418_pos__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ B2 ) ) ) ) ).
% pos_le_divideR_eq
thf(fact_3419_pos__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) @ A3 )
= ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% pos_divideR_le_eq
thf(fact_3420_summable__exp__generic,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( summable @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X3 @ N3 ) ) ) ) ).
% summable_exp_generic
thf(fact_3421_neg__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) @ A3 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divideR_le_eq
thf(fact_3422_neg__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% neg_le_minus_divideR_eq
thf(fact_3423_pos__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) @ A3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% pos_minus_divideR_le_eq
thf(fact_3424_pos__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divideR_eq
thf(fact_3425_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_3426_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M2: nat] :
( ( times_times @ A @ ( power_power @ A @ X3 @ M2 ) @ ( inverse_inverse @ A @ X3 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X3 ) @ ( power_power @ A @ X3 @ M2 ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_3427_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M2: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ X3 @ M2 ) @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ N ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ N ) @ ( power_power @ A @ X3 @ M2 ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_3428_scaleR__left__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,B2: real,X3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A3 @ B2 ) @ X3 )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) ) ) ).
% scaleR_left_distrib
thf(fact_3429_scaleR__left_Oadd,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: real,Y: real,Xa2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ X3 @ Y ) @ Xa2 )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ X3 @ Xa2 ) @ ( real_V8093663219630862766scaleR @ A @ Y @ Xa2 ) ) ) ) ).
% scaleR_left.add
thf(fact_3430_exp__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X3 @ N3 ) )
@ ( exp @ A @ X3 ) ) ) ).
% exp_converges
thf(fact_3431_exp__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X4: A] :
( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X4 @ N3 ) ) ) ) ) ) ).
% exp_def
thf(fact_3432_summable__norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X3 @ N3 ) ) ) ) ) ).
% summable_norm_exp
thf(fact_3433_frac__ge__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X3 ) ) ) ).
% frac_ge_0
thf(fact_3434_frac__1__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) )
= ( archimedean_frac @ A @ X3 ) ) ) ).
% frac_1_eq
thf(fact_3435_inverse__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% inverse_le_imp_le
thf(fact_3436_le__imp__inverse__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% le_imp_inverse_le
thf(fact_3437_inverse__le__imp__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_3438_le__imp__inverse__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_3439_inverse__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X3 ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ) ).
% inverse_le_1_iff
thf(fact_3440_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_3441_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% inverse_add
thf(fact_3442_scaleR__right__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B2: real,A3: real,C3: A] :
( ( ord_less_eq @ real @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ C3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ C3 ) ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_3443_scaleR__right__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B2: real,X3: A] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) ) ) ) ).
% scaleR_right_mono
thf(fact_3444_scaleR__le__cancel__left,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B2 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_3445_scaleR__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_3446_scaleR__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_3447_scaleR__left__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B2: A,A3: A,C3: real] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ real @ C3 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B2 ) ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_3448_scaleR__left__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X3: A,Y: A,A3: real] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y ) ) ) ) ) ).
% scaleR_left_mono
thf(fact_3449_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,E3: A,C3: A,B2: real,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A3 @ B2 ) @ E3 ) @ C3 ) @ D3 ) ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_3450_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,E3: A,C3: A,B2: real,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less_eq @ A @ C3 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ B2 @ A3 ) @ E3 ) @ D3 ) ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_3451_exp__series__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A,Y: A,N: nat] :
( ( ( times_times @ A @ X3 @ Y )
= ( times_times @ A @ Y @ X3 ) )
=> ( ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I4 ) ) @ ( power_power @ A @ X3 @ I4 ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ I4 ) ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% exp_series_add_commuting
thf(fact_3452_exp__first__term,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X4: A] :
( plus_plus @ A @ ( one_one @ A )
@ ( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N3 ) ) ) @ ( power_power @ A @ X4 @ ( suc @ N3 ) ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_3453_inverse__le__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ).
% inverse_le_iff
thf(fact_3454_inverse__less__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ B2 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ) ).
% inverse_less_iff
thf(fact_3455_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_3456_inverse__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ X3 ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
| ( ord_less @ A @ ( one_one @ A ) @ X3 ) ) ) ) ).
% inverse_less_1_iff
thf(fact_3457_one__le__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X3 ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
& ( ord_less_eq @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% one_le_inverse_iff
thf(fact_3458_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ? [N2: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ X3 ) ) ) ).
% reals_Archimedean
thf(fact_3459_scaleR__le__0__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B2: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( A3
= ( zero_zero @ real ) ) ) ) ) ).
% scaleR_le_0_iff
thf(fact_3460_zero__le__scaleR__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ B2 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( A3
= ( zero_zero @ real ) ) ) ) ) ).
% zero_le_scaleR_iff
thf(fact_3461_scaleR__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B2: real,X3: A,Y: A] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ Y ) ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_3462_scaleR__mono_H,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B2: real,C3: A,D3: A] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ( 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 @ B2 @ D3 ) ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_3463_split__scaleR__neg__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X3: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X3 ) @ ( zero_zero @ A ) ) ) ) ).
% split_scaleR_neg_le
thf(fact_3464_split__scaleR__pos__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B2: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ B2 ) ) ) ) ).
% split_scaleR_pos_le
thf(fact_3465_scaleR__nonneg__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X3: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ X3 ) ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_3466_scaleR__nonneg__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X3: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_3467_scaleR__nonpos__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X3: A] :
( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_3468_scaleR__nonpos__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B2: A] :
( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ B2 ) ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_3469_scaleR__left__le__one__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X3: A,A3: real] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ real @ A3 @ ( one_one @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X3 ) @ X3 ) ) ) ) ).
% scaleR_left_le_one_le
thf(fact_3470_scaleR__2,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 )
= ( plus_plus @ A @ X3 @ X3 ) ) ) ).
% scaleR_2
thf(fact_3471_forall__pos__mono__1,axiom,
! [P: real > $o,E3: real] :
( ! [D2: real,E2: real] :
( ( ord_less @ real @ D2 @ E2 )
=> ( ( P @ D2 )
=> ( P @ E2 ) ) )
=> ( ! [N2: nat] : ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( P @ E3 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_3472_exp__first__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [K2: nat] :
( ( exp @ A )
= ( ^ [X4: A] :
( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X4 @ N3 ) )
@ ( set_ord_lessThan @ nat @ K2 ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N3 @ K2 ) ) ) @ ( power_power @ A @ X4 @ ( plus_plus @ nat @ N3 @ K2 ) ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_3473_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X4: int] : X4
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_3474_summable__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N3 ) ) @ ( power_power @ A @ X3 @ N3 ) ) ) ) ).
% summable_exp
thf(fact_3475_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M2: nat,N: nat] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( power_power @ A @ X3 @ ( minus_minus @ nat @ N @ M2 ) )
= ( times_times @ A @ ( power_power @ A @ X3 @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ M2 ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_3476_sin__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ X3 @ N3 ) )
@ ( sin @ A @ X3 ) ) ) ).
% sin_converges
thf(fact_3477_sin__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A )
= ( ^ [X4: A] :
( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ X4 @ N3 ) ) ) ) ) ) ).
% sin_def
thf(fact_3478_cos__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ X3 @ N3 ) )
@ ( cos @ A @ X3 ) ) ) ).
% cos_converges
thf(fact_3479_cos__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cos @ A )
= ( ^ [X4: A] :
( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ X4 @ N3 ) ) ) ) ) ) ).
% cos_def
thf(fact_3480_summable__norm__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) ) ) ) ).
% summable_norm_sin
thf(fact_3481_summable__norm__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) ) ) ) ).
% summable_norm_cos
thf(fact_3482_exp__first__two__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X4: A] :
( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X4 )
@ ( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ A @ X4 @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_3483_frac__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ( archimedean_frac @ A @ X3 )
= X3 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
& ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% frac_eq
thf(fact_3484_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_3485_sin__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( sums @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ N3 ) ) )
@ ( sin @ A @ X3 ) ) ) ).
% sin_minus_converges
thf(fact_3486_cos__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ N3 ) )
@ ( cos @ A @ X3 ) ) ) ).
% cos_minus_converges
thf(fact_3487_exp__plus__inverse__exp,axiom,
! [X3: real] : ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ ( exp @ real @ X3 ) @ ( inverse_inverse @ real @ ( exp @ real @ X3 ) ) ) ) ).
% exp_plus_inverse_exp
thf(fact_3488_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_3489_plus__inverse__ge__2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ X3 @ ( inverse_inverse @ real @ X3 ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_3490_real__inv__sqrt__pow2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( power_power @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( inverse_inverse @ real @ X3 ) ) ) ).
% real_inv_sqrt_pow2
thf(fact_3491_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_3492_tan__cot,axiom,
! [X3: real] :
( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 ) )
= ( inverse_inverse @ real @ ( tan @ real @ X3 ) ) ) ).
% tan_cot
thf(fact_3493_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_3494_real__le__x__sinh,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X3 ) @ ( inverse_inverse @ real @ ( exp @ real @ X3 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_x_sinh
thf(fact_3495_real__le__abs__sinh,axiom,
! [X3: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( abs_abs @ real @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X3 ) @ ( inverse_inverse @ real @ ( exp @ real @ X3 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_abs_sinh
thf(fact_3496_tan__sec,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ ( inverse_inverse @ A @ ( cos @ A @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tan_sec
thf(fact_3497_sinh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( sums @ A
@ ^ [N3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X3 @ N3 ) ) )
@ ( sinh @ A @ X3 ) ) ) ).
% sinh_converges
thf(fact_3498_cosh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( sums @ A
@ ^ [N3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X3 @ N3 ) ) @ ( zero_zero @ A ) )
@ ( cosh @ A @ X3 ) ) ) ).
% cosh_converges
thf(fact_3499_complex__unimodular__polar,axiom,
! [Z2: complex] :
( ( ( real_V7770717601297561774m_norm @ complex @ Z2 )
= ( 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 ) )
=> ( Z2
!= ( complex2 @ ( cos @ real @ T6 ) @ ( sin @ real @ T6 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_3500_arctan__def,axiom,
( arctan
= ( ^ [Y3: real] :
( the @ real
@ ^ [X4: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X4 )
& ( ord_less @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X4 )
= Y3 ) ) ) ) ) ).
% arctan_def
thf(fact_3501_arcsin__def,axiom,
( arcsin
= ( ^ [Y3: real] :
( the @ real
@ ^ [X4: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X4 )
& ( ord_less_eq @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X4 )
= Y3 ) ) ) ) ) ).
% arcsin_def
thf(fact_3502_cosh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( cosh @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cosh @ A @ X3 ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( sinh @ A @ X3 ) @ ( sinh @ A @ Y ) ) ) ) ) ).
% cosh_add
thf(fact_3503_sinh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( sinh @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sinh @ A @ X3 ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( cosh @ A @ X3 ) @ ( sinh @ A @ Y ) ) ) ) ) ).
% sinh_add
thf(fact_3504_cosh__plus__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( ( plus_plus @ A @ ( cosh @ A @ X3 ) @ ( sinh @ A @ X3 ) )
= ( exp @ A @ X3 ) ) ) ).
% cosh_plus_sinh
thf(fact_3505_sinh__plus__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( ( plus_plus @ A @ ( sinh @ A @ X3 ) @ ( cosh @ A @ X3 ) )
= ( exp @ A @ X3 ) ) ) ).
% sinh_plus_cosh
thf(fact_3506_Complex__eq__numeral,axiom,
! [A3: real,B2: real,W: num] :
( ( ( complex2 @ A3 @ B2 )
= ( numeral_numeral @ complex @ W ) )
= ( ( A3
= ( numeral_numeral @ real @ W ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_numeral
thf(fact_3507_sinh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( sinh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sinh @ A @ X3 ) ) @ ( cosh @ A @ X3 ) ) ) ) ).
% sinh_double
thf(fact_3508_Complex__eq__neg__numeral,axiom,
! [A3: real,B2: real,W: num] :
( ( ( complex2 @ A3 @ B2 )
= ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W ) ) )
= ( ( A3
= ( uminus_uminus @ real @ ( numeral_numeral @ real @ W ) ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_neg_numeral
thf(fact_3509_cosh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( power_power @ A @ ( cosh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( power_power @ A @ ( sinh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% cosh_square_eq
thf(fact_3510_hyperbolic__pythagoras,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% hyperbolic_pythagoras
thf(fact_3511_sinh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( power_power @ A @ ( sinh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% sinh_square_eq
thf(fact_3512_complex__inverse,axiom,
! [A3: real,B2: real] :
( ( inverse_inverse @ complex @ ( complex2 @ A3 @ B2 ) )
= ( complex2 @ ( divide_divide @ real @ A3 @ ( plus_plus @ real @ ( power_power @ real @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( uminus_uminus @ real @ B2 ) @ ( plus_plus @ real @ ( power_power @ real @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% complex_inverse
thf(fact_3513_cosh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( cosh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) )
= ( plus_plus @ A @ ( power_power @ A @ ( cosh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_double
thf(fact_3514_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X8: set @ A] :
( the @ A
@ ^ [X4: A] :
( X8
= ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_3515_complex__norm,axiom,
! [X3: real,Y: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( complex2 @ X3 @ Y ) )
= ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_norm
thf(fact_3516_tanh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y: A] :
( ( ( cosh @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cosh @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( tanh @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tanh @ A @ X3 ) @ ( tanh @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tanh @ A @ X3 ) @ ( tanh @ A @ Y ) ) ) ) ) ) ) ) ).
% tanh_add
thf(fact_3517_sinh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( sinh @ A @ X3 )
= ( zero_zero @ A ) )
= ( member @ A @ ( exp @ A @ X3 ) @ ( insert2 @ A @ ( one_one @ A ) @ ( insert2 @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% sinh_zero_iff
thf(fact_3518_cosh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cosh @ A )
= ( ^ [Z4: A] : ( divide_divide @ A @ ( plus_plus @ A @ ( exp @ A @ Z4 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z4 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_field_def
thf(fact_3519_sinh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sinh @ A )
= ( ^ [Z4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z4 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z4 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sinh_field_def
thf(fact_3520_cosh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cosh @ A @ X3 )
= ( zero_zero @ A ) )
= ( ( power_power @ A @ ( exp @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% cosh_zero_iff
thf(fact_3521_pi__half,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
= ( the @ real
@ ^ [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less_eq @ real @ X4 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X4 )
= ( zero_zero @ real ) ) ) ) ) ).
% pi_half
thf(fact_3522_pi__def,axiom,
( pi
= ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) )
@ ( the @ real
@ ^ [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less_eq @ real @ X4 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X4 )
= ( zero_zero @ real ) ) ) ) ) ) ).
% pi_def
thf(fact_3523_cosh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cosh @ A )
= ( ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X4 ) @ ( exp @ A @ ( uminus_uminus @ A @ X4 ) ) ) ) ) ) ) ).
% cosh_def
thf(fact_3524_cosh__ln__real,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( cosh @ real @ ( ln_ln @ real @ X3 ) )
= ( divide_divide @ real @ ( plus_plus @ real @ X3 @ ( inverse_inverse @ real @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% cosh_ln_real
thf(fact_3525_sinh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sinh @ A )
= ( ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ A @ ( exp @ A @ X4 ) @ ( exp @ A @ ( uminus_uminus @ A @ X4 ) ) ) ) ) ) ) ).
% sinh_def
thf(fact_3526_sinh__ln__real,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( sinh @ real @ ( ln_ln @ real @ X3 ) )
= ( divide_divide @ real @ ( minus_minus @ real @ X3 @ ( inverse_inverse @ real @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% sinh_ln_real
thf(fact_3527_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F12: A > B > T,X4: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X4 ) ) ) ) ).
% old.rec_prod_def
thf(fact_3528_powr__int,axiom,
! [X3: real,I: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ I ) )
= ( power_power @ real @ X3 @ ( nat2 @ I ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ I ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( nat2 @ ( uminus_uminus @ int @ I ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_3529_divide__int__unfold,axiom,
! [L: int,K2: int,N: nat,M2: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K2 )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K2 )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K2 )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M2 @ N ) ) ) )
& ( ( ( sgn_sgn @ int @ K2 )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ M2 @ N )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N @ M2 ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_3530_sum__count__set,axiom,
! [A: $tType,Xs2: list @ A,X6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ X6 )
=> ( ( finite_finite2 @ A @ X6 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs2 ) @ X6 )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% sum_count_set
thf(fact_3531_arctan__inverse,axiom,
! [X3: real] :
( ( X3
!= ( zero_zero @ real ) )
=> ( ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ X3 ) )
= ( minus_minus @ real @ ( divide_divide @ real @ ( times_times @ real @ ( sgn_sgn @ real @ X3 ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( arctan @ X3 ) ) ) ) ).
% arctan_inverse
thf(fact_3532_power__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( sgn_sgn @ A @ ( power_power @ A @ A3 @ N ) )
= ( power_power @ A @ ( sgn_sgn @ A @ A3 ) @ N ) ) ) ).
% power_sgn
thf(fact_3533_The__split__eq,axiom,
! [A: $tType,B: $tType,X3: A,Y: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y7: B] :
( ( X3 = X9 )
& ( Y = Y7 ) ) ) )
= ( product_Pair @ A @ B @ X3 @ Y ) ) ).
% The_split_eq
thf(fact_3534_nat__numeral,axiom,
! [K2: num] :
( ( nat2 @ ( numeral_numeral @ int @ K2 ) )
= ( numeral_numeral @ nat @ K2 ) ) ).
% nat_numeral
thf(fact_3535_count__notin,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( count_list @ A @ Xs2 @ X3 )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_3536_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_3537_nat__neg__numeral,axiom,
! [K2: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_3538_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_3539_diff__nat__numeral,axiom,
! [V2: num,V4: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( numeral_numeral @ nat @ V4 ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ V4 ) ) ) ) ).
% diff_nat_numeral
thf(fact_3540_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X3: num,N: nat] :
( ( ( nat2 @ Y )
= ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_3541_numeral__power__eq__nat__cancel__iff,axiom,
! [X3: num,N: nat,Y: int] :
( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N )
= ( nat2 @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N )
= Y ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_3542_nat__ceiling__le__eq,axiom,
! [X3: real,A3: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X3 ) ) @ A3 )
= ( ord_less_eq @ real @ X3 @ ( semiring_1_of_nat @ real @ A3 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_3543_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_3544_nat__numeral__diff__1,axiom,
! [V2: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( one_one @ int ) ) ) ) ).
% nat_numeral_diff_1
thf(fact_3545_numeral__power__less__nat__cancel__iff,axiom,
! [X3: num,N: nat,A3: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N ) @ ( nat2 @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) @ A3 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_3546_nat__less__numeral__power__cancel__iff,axiom,
! [A3: int,X3: num,N: nat] :
( ( ord_less @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_3547_nat__le__numeral__power__cancel__iff,axiom,
! [A3: int,X3: num,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_3548_numeral__power__le__nat__cancel__iff,axiom,
! [X3: num,N: nat,A3: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N ) @ ( nat2 @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) @ A3 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_3549_same__sgn__sgn__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A3: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A3 ) )
=> ( ( sgn_sgn @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( sgn_sgn @ A @ A3 ) ) ) ) ).
% same_sgn_sgn_add
thf(fact_3550_nat__numeral__as__int,axiom,
( ( numeral_numeral @ nat )
= ( ^ [I4: num] : ( nat2 @ ( numeral_numeral @ int @ I4 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_3551_nat__mono,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq @ int @ X3 @ Y )
=> ( ord_less_eq @ nat @ ( nat2 @ X3 ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_3552_same__sgn__abs__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A3: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A3 ) )
=> ( ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% same_sgn_abs_add
thf(fact_3553_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_3554_nat__le__iff,axiom,
! [X3: int,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ X3 ) @ N )
= ( ord_less_eq @ int @ X3 @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% nat_le_iff
thf(fact_3555_nat__int__add,axiom,
! [A3: nat,B2: nat] :
( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) )
= ( plus_plus @ nat @ A3 @ B2 ) ) ).
% nat_int_add
thf(fact_3556_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A8: nat,B8: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A8 ) @ ( semiring_1_of_nat @ int @ B8 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_3557_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_3558_nat__le__eq__zle,axiom,
! [W: int,Z2: int] :
( ( ( ord_less @ int @ ( zero_zero @ int ) @ W )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) )
=> ( ( ord_less_eq @ nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq @ int @ W @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_3559_le__mult__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( nat2 @ ( archim6421214686448440834_floor @ A @ A3 ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ B2 ) ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% le_mult_nat_floor
thf(fact_3560_le__nat__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less_eq @ nat @ N @ ( nat2 @ K2 ) )
= ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K2 ) ) ) ).
% le_nat_iff
thf(fact_3561_nat__add__distrib,axiom,
! [Z2: int,Z6: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
=> ( ( nat2 @ ( plus_plus @ int @ Z2 @ Z6 ) )
= ( plus_plus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_3562_Suc__as__int,axiom,
( suc
= ( ^ [A8: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A8 ) @ ( one_one @ int ) ) ) ) ) ).
% Suc_as_int
thf(fact_3563_nat__abs__triangle__ineq,axiom,
! [K2: int,L: int] : ( ord_less_eq @ nat @ ( nat2 @ ( abs_abs @ int @ ( plus_plus @ int @ K2 @ L ) ) ) @ ( plus_plus @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_3564_nat__power__eq,axiom,
! [Z2: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( nat2 @ ( power_power @ int @ Z2 @ N ) )
= ( power_power @ nat @ ( nat2 @ Z2 ) @ N ) ) ) ).
% nat_power_eq
thf(fact_3565_count__le__length,axiom,
! [A: $tType,Xs2: list @ A,X3: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs2 @ X3 ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% count_le_length
thf(fact_3566_floor__eq3,axiom,
! [N: nat,X3: real] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X3 ) )
= N ) ) ) ).
% floor_eq3
thf(fact_3567_le__nat__floor,axiom,
! [X3: nat,A3: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X3 ) @ A3 )
=> ( ord_less_eq @ nat @ X3 @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A3 ) ) ) ) ).
% le_nat_floor
thf(fact_3568_divide__int__def,axiom,
( ( divide_divide @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( ( sgn_sgn @ int @ K3 )
= ( sgn_sgn @ int @ L2 ) )
@ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) )
@ ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ int @ L2 @ K3 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_3569_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_3570_sgn__power__injE,axiom,
! [A3: real,N: nat,X3: real,B2: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ A3 ) @ ( power_power @ real @ ( abs_abs @ real @ A3 ) @ N ) )
= X3 )
=> ( ( X3
= ( times_times @ real @ ( sgn_sgn @ real @ B2 ) @ ( power_power @ real @ ( abs_abs @ real @ B2 ) @ N ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A3 = B2 ) ) ) ) ).
% sgn_power_injE
thf(fact_3571_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( suc @ ( nat2 @ Z2 ) )
= ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_3572_nat__abs__int__diff,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ B2 @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ A3 @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_3573_floor__eq4,axiom,
! [N: nat,X3: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X3 ) )
= N ) ) ) ).
% floor_eq4
thf(fact_3574_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K2: int,Q3: int] :
( ( ( sgn_sgn @ int @ R2 )
= ( sgn_sgn @ int @ L ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ L ) )
=> ( ( K2
= ( plus_plus @ int @ ( times_times @ int @ Q3 @ L ) @ R2 ) )
=> ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ Q3 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_3575_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A32: product_prod @ int @ int] :
( ? [K3: int] :
( ( A12 = K3 )
& ( A23
= ( zero_zero @ int ) )
& ( A32
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
| ? [L2: int,K3: int,Q4: int] :
( ( A12 = K3 )
& ( A23 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) )
& ( L2
!= ( zero_zero @ int ) )
& ( K3
= ( times_times @ int @ Q4 @ L2 ) ) )
| ? [R5: int,L2: int,K3: int,Q4: int] :
( ( A12 = K3 )
& ( A23 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q4 @ 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 @ Q4 @ L2 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_3576_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A33: product_prod @ int @ int] :
( ( eucl_rel_int @ A1 @ A22 @ A33 )
=> ( ( ( A22
= ( zero_zero @ int ) )
=> ( A33
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A1 ) ) )
=> ( ! [Q2: int] :
( ( A33
= ( product_Pair @ int @ int @ Q2 @ ( zero_zero @ int ) ) )
=> ( ( A22
!= ( zero_zero @ int ) )
=> ( A1
!= ( times_times @ int @ Q2 @ A22 ) ) ) )
=> ~ ! [R3: int,Q2: int] :
( ( A33
= ( product_Pair @ int @ int @ Q2 @ R3 ) )
=> ( ( ( sgn_sgn @ int @ R3 )
= ( sgn_sgn @ int @ A22 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ A22 ) )
=> ( A1
!= ( plus_plus @ int @ ( times_times @ int @ Q2 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_3577_even__nat__iff,axiom,
! [K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat2 @ K2 ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 ) ) ) ).
% even_nat_iff
thf(fact_3578_powr__real__of__int,axiom,
! [X3: real,N: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ N ) )
= ( power_power @ real @ X3 @ ( nat2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ N ) )
= ( inverse_inverse @ real @ ( power_power @ real @ X3 @ ( nat2 @ ( uminus_uminus @ int @ N ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_3579_i__even__power,axiom,
! [N: nat] :
( ( power_power @ complex @ imaginary_unit @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ complex @ ( uminus_uminus @ complex @ ( one_one @ complex ) ) @ N ) ) ).
% i_even_power
thf(fact_3580_power2__i,axiom,
( ( power_power @ complex @ imaginary_unit @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% power2_i
thf(fact_3581_cis__2pi,axiom,
( ( cis @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ complex ) ) ).
% cis_2pi
thf(fact_3582_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_3583_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_3584_divide__numeral__i,axiom,
! [Z2: complex,N: num] :
( ( divide_divide @ complex @ Z2 @ ( times_times @ complex @ ( numeral_numeral @ complex @ N ) @ imaginary_unit ) )
= ( divide_divide @ complex @ ( uminus_uminus @ complex @ ( times_times @ complex @ imaginary_unit @ Z2 ) ) @ ( numeral_numeral @ complex @ N ) ) ) ).
% divide_numeral_i
thf(fact_3585_xor__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% xor_numerals(3)
thf(fact_3586_xor__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% xor_numerals(1)
thf(fact_3587_xor__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit0 @ Y ) ) ) ) ).
% xor_numerals(2)
thf(fact_3588_xor__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X3 ) ) ) ) ).
% xor_numerals(5)
thf(fact_3589_xor__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ X3 ) ) ) ) ).
% xor_numerals(8)
thf(fact_3590_xor__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% xor_numerals(7)
thf(fact_3591_xor__nat__numerals_I4_J,axiom,
! [X3: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) ) ).
% xor_nat_numerals(4)
thf(fact_3592_xor__nat__numerals_I3_J,axiom,
! [X3: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) ) ).
% xor_nat_numerals(3)
thf(fact_3593_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_3594_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_3595_exp__two__pi__i_H,axiom,
( ( exp @ complex @ ( times_times @ complex @ imaginary_unit @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ pi ) @ ( numeral_numeral @ complex @ ( bit0 @ one2 ) ) ) ) )
= ( one_one @ complex ) ) ).
% exp_two_pi_i'
thf(fact_3596_exp__two__pi__i,axiom,
( ( exp @ complex @ ( times_times @ complex @ ( times_times @ complex @ ( numeral_numeral @ complex @ ( bit0 @ one2 ) ) @ ( real_Vector_of_real @ complex @ pi ) ) @ imaginary_unit ) )
= ( one_one @ complex ) ) ).
% exp_two_pi_i
thf(fact_3597_cis__pi__half,axiom,
( ( cis @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= imaginary_unit ) ).
% cis_pi_half
thf(fact_3598_xor__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_3599_xor__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_3600_cis__minus__pi__half,axiom,
( ( cis @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( uminus_uminus @ complex @ imaginary_unit ) ) ).
% cis_minus_pi_half
thf(fact_3601_complex__i__not__numeral,axiom,
! [W: num] :
( imaginary_unit
!= ( numeral_numeral @ complex @ W ) ) ).
% complex_i_not_numeral
thf(fact_3602_complex__i__not__neg__numeral,axiom,
! [W: num] :
( imaginary_unit
!= ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W ) ) ) ).
% complex_i_not_neg_numeral
thf(fact_3603_DeMoivre,axiom,
! [A3: real,N: nat] :
( ( power_power @ complex @ ( cis @ A3 ) @ N )
= ( cis @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A3 ) ) ) ).
% DeMoivre
thf(fact_3604_even__xor__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ A3 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_xor_iff
thf(fact_3605_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M5: nat,N3: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N3
@ ( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ M5
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_3606_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M5: nat,N3: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_3607_xor__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( one_one @ A ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% xor_one_eq
thf(fact_3608_one__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A3 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% one_xor_eq
thf(fact_3609_Arg__minus__ii,axiom,
( ( arg @ ( uminus_uminus @ complex @ imaginary_unit ) )
= ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% Arg_minus_ii
thf(fact_3610_Arg__ii,axiom,
( ( arg @ imaginary_unit )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% Arg_ii
thf(fact_3611_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
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= ( one_one @ complex ) ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_3612_csqrt__ii,axiom,
( ( csqrt @ imaginary_unit )
= ( divide_divide @ complex @ ( plus_plus @ complex @ ( one_one @ complex ) @ imaginary_unit ) @ ( real_Vector_of_real @ complex @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% csqrt_ii
thf(fact_3613_power2__csqrt,axiom,
! [Z2: complex] :
( ( power_power @ complex @ ( csqrt @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= Z2 ) ).
% power2_csqrt
thf(fact_3614_XOR__upper,axiom,
! [X3: int,N: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less @ int @ X3 @ ( 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 @ X3 @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_3615_xor__int__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K3: int,L2: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) )
!= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_3616_infinite__imp__bij__betw,axiom,
! [A: $tType,A6: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ? [H4: A > A] : ( bij_betw @ A @ A @ H4 @ A6 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_3617_infinite__imp__bij__betw2,axiom,
! [A: $tType,A6: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ? [H4: A > A] : ( bij_betw @ A @ A @ H4 @ A6 @ ( sup_sup @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_3618_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ A,C4: set @ B,G3: A > B,B5: set @ A,D4: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A6 @ C4 )
=> ( ( bij_betw @ A @ B @ G3 @ B5 @ D4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A6 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ C4 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ A6 ) @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( sup_sup @ ( set @ A ) @ A6 @ B5 )
@ ( sup_sup @ ( set @ B ) @ C4 @ D4 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_3619_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ A,C4: set @ A,B5: set @ B,D4: set @ B] :
( ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A6 @ C4 ) @ ( sup_sup @ ( set @ B ) @ B5 @ D4 ) )
=> ( ( bij_betw @ A @ B @ F3 @ C4 @ D4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A6 @ C4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ B5 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F3 @ A6 @ B5 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_3620_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X_1: A] : ( P @ ( zero_zero @ nat ) @ X_1 )
=> ( ! [X5: A,N2: nat] :
( ( P @ N2 @ X5 )
=> ? [Y6: A] :
( ( P @ ( suc @ N2 ) @ Y6 )
& ( Q @ N2 @ X5 @ Y6 ) ) )
=> ? [F2: nat > A] :
! [N9: nat] :
( ( P @ N9 @ ( F2 @ N9 ) )
& ( Q @ N9 @ ( F2 @ N9 ) @ ( F2 @ ( suc @ N9 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_3621_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A] :
( ( bij_betw @ A @ B @ F3 @ A6 @ ( bot_bot @ ( set @ B ) ) )
=> ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% bij_betw_empty2
thf(fact_3622_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ B] :
( ( bij_betw @ A @ B @ F3 @ ( bot_bot @ ( set @ A ) ) @ A6 )
=> ( A6
= ( bot_bot @ ( set @ B ) ) ) ) ).
% bij_betw_empty1
thf(fact_3623_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B2: A,A6: set @ A,F3: A > B,A11: set @ B] :
( ~ ( member @ A @ B2 @ A6 )
=> ( ~ ( member @ B @ ( F3 @ B2 ) @ A11 )
=> ( ( bij_betw @ A @ B @ F3 @ A6 @ A11 )
= ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A6 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A11 @ ( insert2 @ B @ ( F3 @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_3624_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B2: A,A6: set @ A,F3: A > B,A11: set @ B] :
( ~ ( member @ A @ B2 @ A6 )
=> ( ~ ( member @ B @ ( F3 @ B2 ) @ A11 )
=> ( ( bij_betw @ A @ B @ F3 @ A6 @ A11 )
=> ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A6 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A11 @ ( insert2 @ B @ ( F3 @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_3625_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ A,B5: set @ B,C4: set @ A,D4: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A6 @ B5 )
=> ( ( bij_betw @ A @ B @ F3 @ C4 @ D4 )
=> ( ( ( inf_inf @ ( set @ B ) @ B5 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A6 @ C4 ) @ ( sup_sup @ ( set @ B ) @ B5 @ D4 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_3626_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
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= ( one_one @ complex ) ) )
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= C3 ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_3627_xor__int__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ L2 )
@ ( if @ int
@ ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ K3 )
@ ( if @ int
@ ( K3
= ( zero_zero @ int ) )
@ L2
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K3
@ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_3628_cis__multiple__2pi,axiom,
! [N: real] :
( ( member @ real @ N @ ( ring_1_Ints @ real ) )
=> ( ( cis @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N ) )
= ( one_one @ complex ) ) ) ).
% cis_multiple_2pi
thf(fact_3629_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_3630_real__root__Suc__0,axiom,
! [X3: real] :
( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X3 )
= X3 ) ).
% real_root_Suc_0
thf(fact_3631_floor__add2,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] :
( ( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
| ( member @ A @ Y @ ( ring_1_Ints @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ) ).
% floor_add2
thf(fact_3632_minus__not__numeral__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( uminus_uminus @ A @ ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ).
% minus_not_numeral_eq
thf(fact_3633_even__not__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_not_iff
thf(fact_3634_not__one__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% not_one_eq
thf(fact_3635_real__root__pow__pos2,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( power_power @ real @ ( root @ N @ X3 ) @ N )
= X3 ) ) ) ).
% real_root_pow_pos2
thf(fact_3636_or__minus__minus__numerals,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% or_minus_minus_numerals
thf(fact_3637_and__minus__minus__numerals,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se1065995026697491101ons_or @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% and_minus_minus_numerals
thf(fact_3638_Ints__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( power_power @ A @ A3 @ N ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_power
thf(fact_3639_Ints__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: num] : ( member @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_numeral
thf(fact_3640_Ints__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_add
thf(fact_3641_of__int__not__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K2: num] :
( ( ring_1_of_int @ A @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ K2 ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ K2 ) ) ) ) ).
% of_int_not_numeral
thf(fact_3642_not__diff__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ B2 ) ) ) ).
% not_diff_distrib
thf(fact_3643_not__add__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( minus_minus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ B2 ) ) ) ).
% not_add_distrib
thf(fact_3644_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_3645_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A3 @ X4 )
& ( ord_less_eq @ A @ X4 @ B2 ) ) ) ) ) ).
% finite_int_segment
thf(fact_3646_real__root__power,axiom,
! [N: nat,X3: real,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X3 @ K2 ) )
= ( power_power @ real @ ( root @ N @ X3 ) @ K2 ) ) ) ).
% real_root_power
thf(fact_3647_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A8: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A8 ) @ ( one_one @ A ) ) ) ) ) ).
% minus_eq_not_plus_1
thf(fact_3648_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_3649_minus__numeral__inc__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% minus_numeral_inc_eq
thf(fact_3650_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_3651_sqrt__def,axiom,
( sqrt
= ( root @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% sqrt_def
thf(fact_3652_not__int__div__2,axiom,
! [K2: int] :
( ( divide_divide @ int @ ( bit_ri4277139882892585799ns_not @ int @ K2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% not_int_div_2
thf(fact_3653_even__not__iff__int,axiom,
! [K2: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ K2 ) )
= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 ) ) ) ).
% even_not_iff_int
thf(fact_3654_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_3655_not__numeral__Bit0__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) ) ) ) ).
% not_numeral_Bit0_eq
thf(fact_3656_and__not__numerals_I4_J,axiom,
! [M2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( numeral_numeral @ int @ ( bit0 @ M2 ) ) ) ).
% and_not_numerals(4)
thf(fact_3657_and__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( one_one @ int ) ) ).
% and_not_numerals(2)
thf(fact_3658_or__not__numerals_I4_J,axiom,
! [M2: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) ) ).
% or_not_numerals(4)
thf(fact_3659_or__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(2)
thf(fact_3660_not__numeral__BitM__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bitM @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) ) ) ) ).
% not_numeral_BitM_eq
thf(fact_3661_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_3662_take__bit__not__mask__eq__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_not_mask_eq_0
thf(fact_3663_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
=> ( ( X3
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X3 ) ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_3664_int__numeral__or__not__num__neg,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M2 @ N ) ) ) ) ).
% int_numeral_or_not_num_neg
thf(fact_3665_int__numeral__not__or__num__neg,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ N @ M2 ) ) ) ) ).
% int_numeral_not_or_num_neg
thf(fact_3666_numeral__or__not__num__eq,axiom,
! [M2: num,N: num] :
( ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M2 @ N ) )
= ( uminus_uminus @ int @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% numeral_or_not_num_eq
thf(fact_3667_real__root__decreasing,axiom,
! [N: nat,N5: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N5 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( root @ N5 @ X3 ) @ ( root @ N @ X3 ) ) ) ) ) ).
% real_root_decreasing
thf(fact_3668_real__root__pow__pos,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( power_power @ real @ ( root @ N @ X3 ) @ N )
= X3 ) ) ) ).
% real_root_pow_pos
thf(fact_3669_real__root__power__cancel,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( root @ N @ ( power_power @ real @ X3 @ N ) )
= X3 ) ) ) ).
% real_root_power_cancel
thf(fact_3670_real__root__pos__unique,axiom,
! [N: nat,Y: real,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( power_power @ real @ Y @ N )
= X3 )
=> ( ( root @ N @ X3 )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_3671_odd__real__root__pow,axiom,
! [N: nat,X3: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ real @ ( root @ N @ X3 ) @ N )
= X3 ) ) ).
% odd_real_root_pow
thf(fact_3672_odd__real__root__unique,axiom,
! [N: nat,Y: real,X3: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ( power_power @ real @ Y @ N )
= X3 )
=> ( ( root @ N @ X3 )
= Y ) ) ) ).
% odd_real_root_unique
thf(fact_3673_odd__real__root__power__cancel,axiom,
! [N: nat,X3: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X3 @ N ) )
= X3 ) ) ).
% odd_real_root_power_cancel
thf(fact_3674_and__not__numerals_I5_J,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% and_not_numerals(5)
thf(fact_3675_and__not__numerals_I7_J,axiom,
! [M2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( numeral_numeral @ int @ ( bit0 @ M2 ) ) ) ).
% and_not_numerals(7)
thf(fact_3676_or__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(3)
thf(fact_3677_and__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( zero_zero @ int ) ) ).
% and_not_numerals(3)
thf(fact_3678_or__not__numerals_I7_J,axiom,
! [M2: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).
% or_not_numerals(7)
thf(fact_3679_real__root__increasing,axiom,
! [N: nat,N5: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N5 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( root @ N @ X3 ) @ ( root @ N5 @ X3 ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_3680_and__not__numerals_I9_J,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% and_not_numerals(9)
thf(fact_3681_and__not__numerals_I6_J,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% and_not_numerals(6)
thf(fact_3682_or__not__numerals_I6_J,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% or_not_numerals(6)
thf(fact_3683_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_3684_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_3685_sgn__power__root,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ real @ ( sgn_sgn @ real @ ( root @ N @ X3 ) ) @ ( power_power @ real @ ( abs_abs @ real @ ( root @ N @ X3 ) ) @ N ) )
= X3 ) ) ).
% sgn_power_root
thf(fact_3686_minus__exp__eq__not__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% minus_exp_eq_not_mask
thf(fact_3687_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A3 ) @ ( archim6421214686448440834_floor @ B @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A3 @ B2 ) ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_3688_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,A3: A] :
( ( ( archimedean_frac @ A @ X3 )
= A3 )
= ( ( member @ A @ ( minus_minus @ A @ X3 @ 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_3689_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A3 @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A3 ) @ ( archimedean_ceiling @ B @ B2 ) ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_3690_or__not__numerals_I5_J,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_3691_split__root,axiom,
! [P: real > $o,N: nat,X3: real] :
( ( P @ ( root @ N @ X3 ) )
= ( ( ( 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 ) )
= X3 )
=> ( P @ Y3 ) ) ) ) ) ).
% split_root
thf(fact_3692_sin__integer__2pi,axiom,
! [N: real] :
( ( member @ real @ N @ ( ring_1_Ints @ real ) )
=> ( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N ) )
= ( zero_zero @ real ) ) ) ).
% sin_integer_2pi
thf(fact_3693_cos__integer__2pi,axiom,
! [N: real] :
( ( member @ real @ N @ ( ring_1_Ints @ real ) )
=> ( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N ) )
= ( one_one @ real ) ) ) ).
% cos_integer_2pi
thf(fact_3694_and__not__numerals_I8_J,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_3695_or__not__numerals_I8_J,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_3696_or__not__numerals_I9_J,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_3697_not__int__rec,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K3: int] : ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_3698_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_3699_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_3700_push__bit__numeral__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_3701_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F6: set @ A,I5: set @ A,F3: A > B,I: A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I5 )
& ( ( F3 @ I4 )
!= ( zero_zero @ B ) ) ) )
@ F6 )
=> ( ( ( 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_3702_Sum__Ico__nat,axiom,
! [M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M2 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Ico_nat
thf(fact_3703_push__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ A3 ) ) ) ).
% push_bit_push_bit
thf(fact_3704_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_3705_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_3706_ivl__subset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,J: A,M2: A,N: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ N ) )
= ( ( ord_less_eq @ A @ J @ I )
| ( ( ord_less_eq @ A @ M2 @ I )
& ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% ivl_subset
thf(fact_3707_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
= ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_3708_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_3709_ivl__diff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,N: A,M2: A] :
( ( ord_less_eq @ A @ I @ N )
=> ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ I @ N ) )
= ( set_or7035219750837199246ssThan @ A @ N @ M2 ) ) ) ) ).
% ivl_diff
thf(fact_3710_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P2: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P2 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_3711_push__bit__Suc__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,K2: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ K2 ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ K2 ) ) ) ) ) ).
% push_bit_Suc_numeral
thf(fact_3712_atLeastLessThan__singleton,axiom,
! [M2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ M2 ) )
= ( insert2 @ nat @ M2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_3713_push__bit__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,K2: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K2 ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K2 ) ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_3714_push__bit__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [L: num,K2: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ K2 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K2 ) ) ) ) ) ).
% push_bit_numeral
thf(fact_3715_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,P2: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( P2 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P2 @ ( insert2 @ B @ I @ I5 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P2 @ I5 ) ) )
& ( ~ ( member @ B @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P2 @ ( insert2 @ B @ I @ I5 ) )
= ( plus_plus @ A @ ( P2 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P2 @ I5 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_3716_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_3717_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ A3 )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_3718_push__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% push_bit_of_1
thf(fact_3719_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_3720_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M2: nat,G3: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G3 @ N ) ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_3721_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G3: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G3 @ N ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_3722_push__bit__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [L: num,K2: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K2 ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K2 ) ) ) ) ) ) ).
% push_bit_minus_numeral
thf(fact_3723_push__bit__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_add
thf(fact_3724_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
| ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_3725_ivl__disj__un__two_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_3726_ivl__disj__int__two_I3_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(3)
thf(fact_3727_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
= ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_3728_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_3729_push__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ ( bit_se4730199178511100633sh_bit @ A @ M2 @ A3 ) ) ) ) ).
% push_bit_take_bit
thf(fact_3730_sum_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_3731_sum_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( plus_plus @ nat @ I4 @ K2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_3732_prod_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_3733_prod_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( plus_plus @ nat @ I4 @ K2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_3734_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I4: B] : ( plus_plus @ A @ ( G3 @ I4 ) @ ( H @ I4 ) )
@ I5 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G3 @ I5 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H @ I5 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_3735_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A3: B,C3: B,B2: B,D3: B,G3: B > A,H: B > A] :
( ( A3 = C3 )
=> ( ( B2 = D3 )
=> ( ! [X5: B] :
( ( ord_less_eq @ B @ C3 @ X5 )
=> ( ( ord_less @ B @ X5 @ D3 )
=> ( ( G3 @ X5 )
= ( H @ X5 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ ( set_or7035219750837199246ssThan @ B @ C3 @ D3 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_3736_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A3: B,C3: B,B2: B,D3: B,G3: B > A,H: B > A] :
( ( A3 = C3 )
=> ( ( B2 = D3 )
=> ( ! [X5: B] :
( ( ord_less_eq @ B @ C3 @ X5 )
=> ( ( ord_less @ B @ X5 @ D3 )
=> ( ( G3 @ X5 )
= ( H @ X5 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H @ ( set_or7035219750837199246ssThan @ B @ C3 @ D3 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_3737_ivl__disj__un__two_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_3738_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_3739_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,P2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P2 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ P2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P2 ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_3740_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,P2: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P2 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ N @ P2 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_3741_ivl__disj__int__two_I7_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(7)
thf(fact_3742_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_3743_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,P2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P2 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ P2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P2 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_3744_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_3745_bit__push__bit__iff__int,axiom,
! [M2: nat,K2: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M2 @ K2 ) @ N )
= ( ( ord_less_eq @ nat @ M2 @ N )
& ( bit_se5641148757651400278ts_bit @ int @ K2 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_3746_bit__push__bit__iff__nat,axiom,
! [M2: nat,Q3: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ Q3 ) @ N )
= ( ( ord_less_eq @ nat @ M2 @ N )
& ( bit_se5641148757651400278ts_bit @ nat @ Q3 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_3747_subset__eq__atLeast0__lessThan__finite,axiom,
! [N5: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N5 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_3748_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T4: set @ B,G3: B > A,H: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X5 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G3 @ X5 )
= ( H @ X5 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ T4 )
= ( groups1027152243600224163dd_sum @ B @ A @ H @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_3749_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T4: set @ B,H: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( H @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G3 @ X5 )
= ( H @ X5 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ H @ T4 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_3750_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T4: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ T4 )
= ( groups1027152243600224163dd_sum @ B @ A @ G3 @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_3751_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T4: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ G3 @ T4 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_3752_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( set_or7035219750837199246ssThan @ nat @ I @ ( plus_plus @ nat @ J @ K2 ) )
= ( sup_sup @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) @ ( set_or7035219750837199246ssThan @ nat @ J @ ( plus_plus @ nat @ J @ K2 ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_3753_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_3754_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_3755_ivl__disj__un__two__touch_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_3756_sum__shift__lb__Suc0__0__upt,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,K2: nat] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_3757_sum_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G3 @ N ) ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_3758_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( G3 @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_3759_sum_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat,B2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G3 @ B2 ) ) ) ) ) ).
% sum.atLeastLessThan_Suc
thf(fact_3760_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G3 @ N ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_3761_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( G3 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( H @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I4: B] : ( plus_plus @ A @ ( G3 @ I4 ) @ ( H @ I4 ) )
@ I5 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G3 @ I5 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H @ I5 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_3762_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A8: A,B8: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A8 @ B8 ) @ ( insert2 @ A @ B8 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_3763_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G3 @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_3764_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat,B2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G3 @ B2 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_3765_sum_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( G3 @ N ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum.last_plus
thf(fact_3766_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G3 @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_3767_push__bit__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% push_bit_double
thf(fact_3768_atLeastLessThanSuc,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) )
= ( insert2 @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_3769_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I4 ) ) @ ( F3 @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F3 @ N ) @ ( F3 @ M2 ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_3770_sum_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_3771_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_3772_prod_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat,M2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_3773_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_3774_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M5: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M5 @ K2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ M5 @ K2 ) @ K2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K2 ) ) ) ) ) ).
% sum.nat_group
thf(fact_3775_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,K2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M5: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M5 @ K2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ M5 @ K2 ) @ K2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K2 ) ) ) ) ) ).
% prod.nat_group
thf(fact_3776_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_3777_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_3778_push__bit__mask__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se2239418461657761734s_mask @ A @ N ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ N @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ M2 ) ) ) ) ) ).
% push_bit_mask_eq
thf(fact_3779_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_3780_sum_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M2: nat,G3: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G3 @ N ) ) ) ) ) ) ).
% sum.head_if
thf(fact_3781_push__bit__nat__def,axiom,
( ( bit_se4730199178511100633sh_bit @ nat )
= ( ^ [N3: nat,M5: nat] : ( times_times @ nat @ M5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% push_bit_nat_def
thf(fact_3782_push__bit__int__def,axiom,
( ( bit_se4730199178511100633sh_bit @ int )
= ( ^ [N3: nat,K3: int] : ( times_times @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% push_bit_int_def
thf(fact_3783_push__bit__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A )
= ( ^ [N3: nat,A8: A] : ( times_times @ A @ A8 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% push_bit_eq_mult
thf(fact_3784_exp__dvdE,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A3 )
=> ~ ! [B4: A] :
( A3
!= ( bit_se4730199178511100633sh_bit @ A @ N @ B4 ) ) ) ) ).
% exp_dvdE
thf(fact_3785_fact__prod__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ) ).
% fact_prod_Suc
thf(fact_3786_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_3787_atLeastLessThan__nat__numeral,axiom,
! [M2: nat,K2: num] :
( ( ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K2 ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K2 ) )
= ( insert2 @ nat @ ( pred_numeral @ K2 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( pred_numeral @ K2 ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K2 ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K2 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_3788_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat,M2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_3789_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A8: A,N3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A8 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_3790_summable__Cauchy,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ( ( summable @ A )
= ( ^ [F4: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N6: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ N6 @ M5 )
=> ! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F4 @ ( set_or7035219750837199246ssThan @ nat @ M5 @ N3 ) ) ) @ E4 ) ) ) ) ) ) ).
% summable_Cauchy
thf(fact_3791_push__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% push_bit_minus_one
thf(fact_3792_sums__group,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A,S: A,K2: nat] :
( ( sums @ A @ F3 @ S )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( sums @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N3 @ K2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ K2 ) @ K2 ) ) )
@ S ) ) ) ) ).
% sums_group
thf(fact_3793_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_3794_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N @ K2 ) @ N ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ) ).
% fact_split
thf(fact_3795_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K2 @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_3796_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_3797_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N3: nat,A8: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A8 ) @ N3 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A8 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N3 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A8 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_3798_sum__power2,axiom,
! [K2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) )
= ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 ) @ ( one_one @ nat ) ) ) ).
% sum_power2
thf(fact_3799_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A8: A,Xs: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F4 @ ( nth @ B @ Xs @ N3 ) ) @ ( power_power @ A @ A8 @ N3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_3800_Chebyshev__sum__upper,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: nat > A,B2: nat > A] :
( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( A3 @ I3 ) @ ( A3 @ J2 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( B2 @ J2 ) @ ( B2 @ I3 ) ) ) )
=> ( ord_less_eq @ A
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A3 @ K3 ) @ ( B2 @ K3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_3801_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A3: nat > nat,B2: nat > nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( A3 @ I3 ) @ ( A3 @ J2 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( B2 @ J2 ) @ ( B2 @ I3 ) ) ) )
=> ( ord_less_eq @ nat
@ ( times_times @ nat @ N
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ ( A3 @ I4 ) @ ( B2 @ 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 @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_3802_Cauchy__iff2,axiom,
( ( topolo3814608138187158403Cauchy @ real )
= ( ^ [X8: nat > real] :
! [J3: nat] :
? [M9: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M9 @ M5 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M9 @ N3 )
=> ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_3803_Cauchy__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M9: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M9 @ M5 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M9 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_3804_CauchyI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M10: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ M10 @ M )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M10 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ M ) @ ( X6 @ N2 ) ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X6 ) ) ) ).
% CauchyI
thf(fact_3805_CauchyD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A,E3: real] :
( ( topolo3814608138187158403Cauchy @ A @ X6 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M8: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M8 @ M3 )
=> ! [N9: nat] :
( ( ord_less_eq @ nat @ M8 @ N9 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ M3 ) @ ( X6 @ N9 ) ) ) @ E3 ) ) ) ) ) ) ).
% CauchyD
thf(fact_3806_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_3807_length__subseqs,axiom,
! [A: $tType,Xs2: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( subseqs @ A @ Xs2 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_subseqs
thf(fact_3808_Code__Target__Int_Opositive__def,axiom,
( code_Target_positive
= ( numeral_numeral @ int ) ) ).
% Code_Target_Int.positive_def
thf(fact_3809_csqrt_Osimps_I1_J,axiom,
! [Z2: complex] :
( ( re @ ( csqrt @ Z2 ) )
= ( sqrt @ ( divide_divide @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% csqrt.simps(1)
thf(fact_3810_complex__Re__numeral,axiom,
! [V2: num] :
( ( re @ ( numeral_numeral @ complex @ V2 ) )
= ( numeral_numeral @ real @ V2 ) ) ).
% complex_Re_numeral
thf(fact_3811_Re__divide__numeral,axiom,
! [Z2: complex,W: num] :
( ( re @ ( divide_divide @ complex @ Z2 @ ( numeral_numeral @ complex @ W ) ) )
= ( divide_divide @ real @ ( re @ Z2 ) @ ( numeral_numeral @ real @ W ) ) ) ).
% Re_divide_numeral
thf(fact_3812_size__list__estimation,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Y: nat,F3: A > nat] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( ord_less @ nat @ Y @ ( F3 @ X3 ) )
=> ( ord_less @ nat @ Y @ ( size_list @ A @ F3 @ Xs2 ) ) ) ) ).
% size_list_estimation
thf(fact_3813_size__list__estimation_H,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Y: nat,F3: A > nat] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ Y @ ( F3 @ X3 ) )
=> ( ord_less_eq @ nat @ Y @ ( size_list @ A @ F3 @ Xs2 ) ) ) ) ).
% size_list_estimation'
thf(fact_3814_size__list__pointwise,axiom,
! [A: $tType,Xs2: list @ A,F3: A > nat,G3: A > nat] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ nat @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) )
=> ( ord_less_eq @ nat @ ( size_list @ A @ F3 @ Xs2 ) @ ( size_list @ A @ G3 @ Xs2 ) ) ) ).
% size_list_pointwise
thf(fact_3815_cos__n__Re__cis__pow__n,axiom,
! [N: nat,A3: real] :
( ( cos @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A3 ) )
= ( re @ ( power_power @ complex @ ( cis @ A3 ) @ N ) ) ) ).
% cos_n_Re_cis_pow_n
thf(fact_3816_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_3817_csqrt_Ocode,axiom,
( csqrt
= ( ^ [Z4: complex] :
( complex2 @ ( sqrt @ ( divide_divide @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z4 ) @ ( re @ Z4 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( times_times @ real
@ ( if @ real
@ ( ( im @ Z4 )
= ( zero_zero @ real ) )
@ ( one_one @ real )
@ ( sgn_sgn @ real @ ( im @ Z4 ) ) )
@ ( sqrt @ ( divide_divide @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z4 ) @ ( re @ Z4 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% csqrt.code
thf(fact_3818_csqrt_Osimps_I2_J,axiom,
! [Z2: complex] :
( ( im @ ( csqrt @ Z2 ) )
= ( times_times @ real
@ ( if @ real
@ ( ( im @ Z2 )
= ( zero_zero @ real ) )
@ ( one_one @ real )
@ ( sgn_sgn @ real @ ( im @ Z2 ) ) )
@ ( sqrt @ ( divide_divide @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% csqrt.simps(2)
thf(fact_3819_Complex__divide,axiom,
( ( divide_divide @ complex )
= ( ^ [X4: complex,Y3: complex] : ( complex2 @ ( divide_divide @ real @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X4 ) @ ( re @ Y3 ) ) @ ( times_times @ real @ ( im @ X4 ) @ ( im @ Y3 ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ ( times_times @ real @ ( im @ X4 ) @ ( re @ Y3 ) ) @ ( times_times @ real @ ( re @ X4 ) @ ( im @ Y3 ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% Complex_divide
thf(fact_3820_Im__power__real,axiom,
! [X3: complex,N: nat] :
( ( ( im @ X3 )
= ( zero_zero @ real ) )
=> ( ( im @ ( power_power @ complex @ X3 @ N ) )
= ( zero_zero @ real ) ) ) ).
% Im_power_real
thf(fact_3821_complex__Im__numeral,axiom,
! [V2: num] :
( ( im @ ( numeral_numeral @ complex @ V2 ) )
= ( zero_zero @ real ) ) ).
% complex_Im_numeral
thf(fact_3822_Re__power__real,axiom,
! [X3: complex,N: nat] :
( ( ( im @ X3 )
= ( zero_zero @ real ) )
=> ( ( re @ ( power_power @ complex @ X3 @ N ) )
= ( power_power @ real @ ( re @ X3 ) @ N ) ) ) ).
% Re_power_real
thf(fact_3823_Im__divide__numeral,axiom,
! [Z2: complex,W: num] :
( ( im @ ( divide_divide @ complex @ Z2 @ ( numeral_numeral @ complex @ W ) ) )
= ( divide_divide @ real @ ( im @ Z2 ) @ ( numeral_numeral @ real @ W ) ) ) ).
% Im_divide_numeral
thf(fact_3824_sin__n__Im__cis__pow__n,axiom,
! [N: nat,A3: real] :
( ( sin @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A3 ) )
= ( im @ ( power_power @ complex @ ( cis @ A3 ) @ N ) ) ) ).
% sin_n_Im_cis_pow_n
thf(fact_3825_VEBT_Osize__gen_I2_J,axiom,
! [X21: $o,X222: $o] :
( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size_gen(2)
thf(fact_3826_cmod__power2,axiom,
! [Z2: complex] :
( ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ real @ ( power_power @ real @ ( re @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% cmod_power2
thf(fact_3827_Im__power2,axiom,
! [X3: complex] :
( ( im @ ( power_power @ complex @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ X3 ) ) @ ( im @ X3 ) ) ) ).
% Im_power2
thf(fact_3828_Re__power2,axiom,
! [X3: complex] :
( ( re @ ( power_power @ complex @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ real @ ( power_power @ real @ ( re @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% Re_power2
thf(fact_3829_complex__eq__0,axiom,
! [Z2: complex] :
( ( Z2
= ( zero_zero @ complex ) )
= ( ( plus_plus @ real @ ( power_power @ real @ ( re @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ) ).
% complex_eq_0
thf(fact_3830_norm__complex__def,axiom,
( ( real_V7770717601297561774m_norm @ complex )
= ( ^ [Z4: complex] : ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% norm_complex_def
thf(fact_3831_inverse__complex_Osimps_I1_J,axiom,
! [X3: complex] :
( ( re @ ( inverse_inverse @ complex @ X3 ) )
= ( divide_divide @ real @ ( re @ X3 ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% inverse_complex.simps(1)
thf(fact_3832_complex__neq__0,axiom,
! [Z2: complex] :
( ( Z2
!= ( zero_zero @ complex ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_neq_0
thf(fact_3833_Re__divide,axiom,
! [X3: complex,Y: complex] :
( ( re @ ( divide_divide @ complex @ X3 @ Y ) )
= ( divide_divide @ real @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X3 ) @ ( re @ Y ) ) @ ( times_times @ real @ ( im @ X3 ) @ ( im @ Y ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Re_divide
thf(fact_3834_csqrt__unique,axiom,
! [W: complex,Z2: complex] :
( ( ( power_power @ complex @ W @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= Z2 )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ W ) )
| ( ( ( re @ W )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ W ) ) ) )
=> ( ( csqrt @ Z2 )
= W ) ) ) ).
% csqrt_unique
thf(fact_3835_csqrt__square,axiom,
! [B2: complex] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ B2 ) )
| ( ( ( re @ B2 )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ B2 ) ) ) )
=> ( ( csqrt @ ( power_power @ complex @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= B2 ) ) ).
% csqrt_square
thf(fact_3836_inverse__complex_Osimps_I2_J,axiom,
! [X3: complex] :
( ( im @ ( inverse_inverse @ complex @ X3 ) )
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( im @ X3 ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% inverse_complex.simps(2)
thf(fact_3837_Im__divide,axiom,
! [X3: complex,Y: complex] :
( ( im @ ( divide_divide @ complex @ X3 @ Y ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( times_times @ real @ ( im @ X3 ) @ ( re @ Y ) ) @ ( times_times @ real @ ( re @ X3 ) @ ( im @ Y ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Im_divide
thf(fact_3838_complex__abs__le__norm,axiom,
! [Z2: complex] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( abs_abs @ real @ ( re @ Z2 ) ) @ ( abs_abs @ real @ ( im @ Z2 ) ) ) @ ( times_times @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) ) ) ).
% complex_abs_le_norm
thf(fact_3839_complex__unit__circle,axiom,
! [Z2: complex] :
( ( Z2
!= ( zero_zero @ complex ) )
=> ( ( plus_plus @ real @ ( power_power @ real @ ( divide_divide @ real @ ( re @ Z2 ) @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( divide_divide @ real @ ( im @ Z2 ) @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ) ).
% complex_unit_circle
thf(fact_3840_inverse__complex_Ocode,axiom,
( ( inverse_inverse @ complex )
= ( ^ [X4: complex] : ( complex2 @ ( divide_divide @ real @ ( re @ X4 ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( uminus_uminus @ real @ ( im @ X4 ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% inverse_complex.code
thf(fact_3841_Im__Reals__divide,axiom,
! [R2: complex,Z2: complex] :
( ( member @ complex @ R2 @ ( real_Vector_Reals @ complex ) )
=> ( ( im @ ( divide_divide @ complex @ R2 @ Z2 ) )
= ( divide_divide @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( re @ R2 ) ) @ ( im @ Z2 ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Im_Reals_divide
thf(fact_3842_length__mul__elem,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),N: nat] :
( ! [X5: list @ A] :
( ( member @ ( list @ A ) @ X5 @ ( set2 @ ( list @ A ) @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ X5 )
= N ) )
=> ( ( size_size @ ( list @ A ) @ ( concat @ A @ Xs2 ) )
= ( times_times @ nat @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) @ N ) ) ) ).
% length_mul_elem
thf(fact_3843_Re__Reals__divide,axiom,
! [R2: complex,Z2: complex] :
( ( member @ complex @ R2 @ ( real_Vector_Reals @ complex ) )
=> ( ( re @ ( divide_divide @ complex @ R2 @ Z2 ) )
= ( divide_divide @ real @ ( times_times @ real @ ( re @ R2 ) @ ( re @ Z2 ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Re_Reals_divide
thf(fact_3844_complex__mult__cnj,axiom,
! [Z2: complex] :
( ( times_times @ complex @ Z2 @ ( cnj @ Z2 ) )
= ( real_Vector_of_real @ complex @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_mult_cnj
thf(fact_3845_complex__cnj__power,axiom,
! [X3: complex,N: nat] :
( ( cnj @ ( power_power @ complex @ X3 @ N ) )
= ( power_power @ complex @ ( cnj @ X3 ) @ N ) ) ).
% complex_cnj_power
thf(fact_3846_complex__cnj__numeral,axiom,
! [W: num] :
( ( cnj @ ( numeral_numeral @ complex @ W ) )
= ( numeral_numeral @ complex @ W ) ) ).
% complex_cnj_numeral
thf(fact_3847_complex__cnj__neg__numeral,axiom,
! [W: num] :
( ( cnj @ ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W ) ) )
= ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W ) ) ) ).
% complex_cnj_neg_numeral
thf(fact_3848_Reals__add,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( real_Vector_Reals @ A ) )
=> ( ( member @ A @ B2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ).
% Reals_add
thf(fact_3849_Reals__numeral,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [W: num] : ( member @ A @ ( numeral_numeral @ A @ W ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_numeral
thf(fact_3850_Reals__power,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [A3: A,N: nat] :
( ( member @ A @ A3 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( power_power @ A @ A3 @ N ) @ ( real_Vector_Reals @ A ) ) ) ) ).
% Reals_power
thf(fact_3851_complex__mod__mult__cnj,axiom,
! [Z2: complex] :
( ( real_V7770717601297561774m_norm @ complex @ ( times_times @ complex @ Z2 @ ( cnj @ Z2 ) ) )
= ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% complex_mod_mult_cnj
thf(fact_3852_complex__norm__square,axiom,
! [Z2: complex] :
( ( real_Vector_of_real @ complex @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( times_times @ complex @ Z2 @ ( cnj @ Z2 ) ) ) ).
% complex_norm_square
thf(fact_3853_complex__add__cnj,axiom,
! [Z2: complex] :
( ( plus_plus @ complex @ Z2 @ ( cnj @ Z2 ) )
= ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ Z2 ) ) ) ) ).
% complex_add_cnj
thf(fact_3854_series__comparison__complex,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G3: nat > complex,N5: nat,F3: nat > A] :
( ( summable @ complex @ G3 )
=> ( ! [N2: nat] : ( member @ complex @ ( G3 @ N2 ) @ ( real_Vector_Reals @ complex ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( G3 @ N2 ) ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ ( G3 @ N2 ) ) ) )
=> ( summable @ A @ F3 ) ) ) ) ) ) ).
% series_comparison_complex
thf(fact_3855_complex__diff__cnj,axiom,
! [Z2: complex] :
( ( minus_minus @ complex @ Z2 @ ( cnj @ Z2 ) )
= ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( im @ Z2 ) ) ) @ imaginary_unit ) ) ).
% complex_diff_cnj
thf(fact_3856_complex__div__cnj,axiom,
( ( divide_divide @ complex )
= ( ^ [A8: complex,B8: complex] : ( divide_divide @ complex @ ( times_times @ complex @ A8 @ ( cnj @ B8 ) ) @ ( real_Vector_of_real @ complex @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ B8 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% complex_div_cnj
thf(fact_3857_cnj__add__mult__eq__Re,axiom,
! [Z2: complex,W: complex] :
( ( plus_plus @ complex @ ( times_times @ complex @ Z2 @ ( cnj @ W ) ) @ ( times_times @ complex @ ( cnj @ Z2 ) @ W ) )
= ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ ( times_times @ complex @ Z2 @ ( cnj @ W ) ) ) ) ) ) ).
% cnj_add_mult_eq_Re
thf(fact_3858_set__n__lists,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs2 ) )
= ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys3 )
= N )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_3859_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L2: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q4: code_integer,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 ) ) @ Q4 ) @ ( 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 ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_3860_even__sum__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( finite_card @ B
@ ( collect @ B
@ ^ [A8: B] :
( ( member @ B @ A8 @ A6 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F3 @ A8 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_3861_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_3862_card__atMost,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_3863_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_3864_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_3865_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) )
= ( minus_minus @ nat @ ( suc @ U ) @ L ) ) ).
% card_atLeastAtMost
thf(fact_3866_prod__constant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: A,A6: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : Y
@ A6 )
= ( power_power @ A @ Y @ ( finite_card @ B @ A6 ) ) ) ) ).
% prod_constant
thf(fact_3867_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_3868_card__0__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( finite_card @ A @ A6 )
= ( zero_zero @ nat ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_3869_card__insert__disjoint,axiom,
! [A: $tType,A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X3 @ A6 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( suc @ ( finite_card @ A @ A6 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_3870_card__Diff__insert,axiom,
! [A: $tType,A3: A,A6: set @ A,B5: set @ A] :
( ( member @ A @ A3 @ A6 )
=> ( ~ ( member @ A @ A3 @ B5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ B5 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_3871_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F1: A,F22: nat > A,Nat: nat] :
( ( H @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( case_nat @ B @ ( H @ F1 )
@ ^ [X4: nat] : ( H @ ( F22 @ X4 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_3872_divmod__integer_H__def,axiom,
( ( unique8689654367752047608divmod @ code_integer )
= ( ^ [M5: num,N3: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M5 ) @ ( numeral_numeral @ code_integer @ N3 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M5 ) @ ( numeral_numeral @ code_integer @ N3 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_3873_n__subsets,axiom,
! [A: $tType,A6: set @ A,K2: nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_card @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
& ( ( finite_card @ A @ B6 )
= K2 ) ) ) )
= ( binomial @ ( finite_card @ A @ A6 ) @ K2 ) ) ) ).
% n_subsets
thf(fact_3874_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F22: nat > A,X2: nat] :
( ( case_nat @ A @ F1 @ F22 @ ( suc @ X2 ) )
= ( F22 @ X2 ) ) ).
% old.nat.simps(5)
thf(fact_3875_old_Onat_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F22: nat > A] :
( ( case_nat @ A @ F1 @ F22 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(4)
thf(fact_3876_card__subset__eq,axiom,
! [A: $tType,B5: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( ( finite_card @ A @ A6 )
= ( finite_card @ A @ B5 ) )
=> ( A6 = B5 ) ) ) ) ).
% card_subset_eq
thf(fact_3877_infinite__arbitrarily__large,axiom,
! [A: $tType,A6: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ A6 )
=> ? [B7: set @ A] :
( ( finite_finite2 @ A @ B7 )
& ( ( finite_card @ A @ B7 )
= N )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_3878_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B5: set @ A,A6: set @ B,R2: B > A > $o] :
( ( finite_finite2 @ A @ B5 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ A6 )
=> ? [B10: A] :
( ( member @ A @ B10 @ B5 )
& ( R2 @ A5 @ B10 ) ) )
=> ( ! [A13: B,A24: B,B4: A] :
( ( member @ B @ A13 @ A6 )
=> ( ( member @ B @ A24 @ A6 )
=> ( ( member @ A @ B4 @ B5 )
=> ( ( R2 @ A13 @ B4 )
=> ( ( R2 @ A24 @ B4 )
=> ( A13 = A24 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A6 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_3879_card__insert__le,axiom,
! [A: $tType,A6: set @ A,X3: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ ( insert2 @ A @ X3 @ A6 ) ) ) ).
% card_insert_le
thf(fact_3880_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_3881_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_3882_card__lists__length__eq,axiom,
! [A: $tType,A6: set @ A,N: nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A6 )
& ( ( size_size @ ( list @ A ) @ Xs )
= N ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A6 ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_3883_is__singleton__altdef,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
( ( finite_card @ A @ A7 )
= ( one_one @ nat ) ) ) ) ).
% is_singleton_altdef
thf(fact_3884_card__2__iff_H,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ S3 )
& ? [Y3: A] :
( ( member @ A @ Y3 @ S3 )
& ( X4 != Y3 )
& ! [Z4: A] :
( ( member @ A @ Z4 @ S3 )
=> ( ( Z4 = X4 )
| ( Z4 = Y3 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_3885_card__eq__0__iff,axiom,
! [A: $tType,A6: set @ A] :
( ( ( finite_card @ A @ A6 )
= ( zero_zero @ nat ) )
= ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A6 ) ) ) ).
% card_eq_0_iff
thf(fact_3886_card__Suc__eq__finite,axiom,
! [A: $tType,A6: set @ A,K2: nat] :
( ( ( finite_card @ A @ A6 )
= ( suc @ K2 ) )
= ( ? [B8: A,B6: set @ A] :
( ( A6
= ( insert2 @ A @ B8 @ B6 ) )
& ~ ( member @ A @ B8 @ B6 )
& ( ( finite_card @ A @ B6 )
= K2 )
& ( finite_finite2 @ A @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_3887_card__insert__if,axiom,
! [A: $tType,A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( member @ A @ X3 @ A6 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( finite_card @ A @ A6 ) ) )
& ( ~ ( member @ A @ X3 @ A6 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( suc @ ( finite_card @ A @ A6 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_3888_obtain__subset__with__card__n,axiom,
! [A: $tType,N: nat,S3: set @ A] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S3 ) )
=> ~ ! [T5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T5 @ S3 )
=> ( ( ( finite_card @ A @ T5 )
= N )
=> ~ ( finite_finite2 @ A @ T5 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_3889_card__mono,axiom,
! [A: $tType,B5: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B5 ) ) ) ) ).
% card_mono
thf(fact_3890_card__seteq,axiom,
! [A: $tType,B5: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B5 ) @ ( finite_card @ A @ A6 ) )
=> ( A6 = B5 ) ) ) ) ).
% card_seteq
thf(fact_3891_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F6: set @ A,C4: nat] :
( ! [G5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G5 @ F6 )
=> ( ( finite_finite2 @ A @ G5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G5 ) @ C4 ) ) )
=> ( ( finite_finite2 @ A @ F6 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F6 ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_3892_card__le__sym__Diff,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B5 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B5 @ A6 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_3893_card__length,axiom,
! [A: $tType,Xs2: list @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( set2 @ A @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% card_length
thf(fact_3894_card__1__singletonE,axiom,
! [A: $tType,A6: set @ A] :
( ( ( finite_card @ A @ A6 )
= ( one_one @ nat ) )
=> ~ ! [X5: A] :
( A6
!= ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_3895_card__Un__le,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B5 ) ) ) ).
% card_Un_le
thf(fact_3896_card__less__Suc2,axiom,
! [M7: set @ nat,I: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M7 )
& ( ord_less @ nat @ K3 @ I ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_3897_card__less__Suc,axiom,
! [M7: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M7 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_3898_card__less,axiom,
! [M7: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_3899_subset__card__intvl__is__intvl,axiom,
! [A6: set @ nat,K2: nat] :
( ( ord_less_eq @ ( set @ nat ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ K2 @ ( plus_plus @ nat @ K2 @ ( finite_card @ nat @ A6 ) ) ) )
=> ( A6
= ( set_or7035219750837199246ssThan @ nat @ K2 @ ( plus_plus @ nat @ K2 @ ( finite_card @ nat @ A6 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_3900_sum__Suc,axiom,
! [A: $tType,F3: A > nat,A6: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( suc @ ( F3 @ X4 ) )
@ A6 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A6 ) @ ( finite_card @ A @ A6 ) ) ) ).
% sum_Suc
thf(fact_3901_less__eq__nat_Osimps_I2_J,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
= ( case_nat @ $o @ $false @ ( ord_less_eq @ nat @ M2 ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_3902_max__Suc1,axiom,
! [N: nat,M2: nat] :
( ( ord_max @ nat @ ( suc @ N ) @ M2 )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M6: nat] : ( suc @ ( ord_max @ nat @ N @ M6 ) )
@ M2 ) ) ).
% max_Suc1
thf(fact_3903_max__Suc2,axiom,
! [M2: nat,N: nat] :
( ( ord_max @ nat @ M2 @ ( suc @ N ) )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M6: nat] : ( suc @ ( ord_max @ nat @ M6 @ N ) )
@ M2 ) ) ).
% max_Suc2
thf(fact_3904_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A6: set @ B,K5: A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ord_less_eq @ A @ K5 @ ( F3 @ I3 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) ) ) ) ).
% sum_bounded_below
thf(fact_3905_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A6: set @ B,F3: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ K5 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above
thf(fact_3906_card__gt__0__iff,axiom,
! [A: $tType,A6: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A6 ) )
= ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A6 ) ) ) ).
% card_gt_0_iff
thf(fact_3907_card__Suc__eq,axiom,
! [A: $tType,A6: set @ A,K2: nat] :
( ( ( finite_card @ A @ A6 )
= ( suc @ K2 ) )
= ( ? [B8: A,B6: set @ A] :
( ( A6
= ( insert2 @ A @ B8 @ B6 ) )
& ~ ( member @ A @ B8 @ B6 )
& ( ( finite_card @ A @ B6 )
= K2 )
& ( ( K2
= ( zero_zero @ nat ) )
=> ( B6
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_3908_card__eq__SucD,axiom,
! [A: $tType,A6: set @ A,K2: nat] :
( ( ( finite_card @ A @ A6 )
= ( suc @ K2 ) )
=> ? [B4: A,B7: set @ A] :
( ( A6
= ( insert2 @ A @ B4 @ B7 ) )
& ~ ( member @ A @ B4 @ B7 )
& ( ( finite_card @ A @ B7 )
= K2 )
& ( ( K2
= ( zero_zero @ nat ) )
=> ( B7
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_3909_card__1__singleton__iff,axiom,
! [A: $tType,A6: set @ A] :
( ( ( finite_card @ A @ A6 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X4: A] :
( A6
= ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_3910_card__le__Suc0__iff__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( X4 = Y3 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_3911_card__le__Suc__iff,axiom,
! [A: $tType,N: nat,A6: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A6 ) )
= ( ? [A8: A,B6: set @ A] :
( ( A6
= ( insert2 @ A @ A8 @ B6 ) )
& ~ ( member @ A @ A8 @ B6 )
& ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B6 ) )
& ( finite_finite2 @ A @ B6 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_3912_card__Diff1__le,axiom,
! [A: $tType,A6: set @ A,X3: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) ) ).
% card_Diff1_le
thf(fact_3913_card__Diff__subset,axiom,
! [A: $tType,B5: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_3914_card__psubset,axiom,
! [A: $tType,B5: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B5 ) )
=> ( ord_less @ ( set @ A ) @ A6 @ B5 ) ) ) ) ).
% card_psubset
thf(fact_3915_diff__card__le__card__Diff,axiom,
! [A: $tType,B5: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B5 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_3916_card__Un__Int,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( plus_plus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B5 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_3917_card__lists__length__le,axiom,
! [A: $tType,A6: set @ A,N: nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A6 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A6 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_3918_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
@ ^ [Z4: A] :
( ( power_power @ A @ Z4 @ N )
= ( one_one @ A ) ) ) )
@ N ) ) ) ).
% card_roots_unity
thf(fact_3919_subset__eq__atLeast0__lessThan__card,axiom,
! [N5: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N5 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_3920_card__sum__le__nat__sum,axiom,
! [S3: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S3 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ S3 ) ) ).
% card_sum_le_nat_sum
thf(fact_3921_card__nth__roots,axiom,
! [C3: complex,N: nat] :
( ( C3
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= C3 ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_3922_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= ( one_one @ complex ) ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_3923_length__n__lists__elem,axiom,
! [A: $tType,Ys: list @ A,N: nat,Xs2: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs2 ) ) )
=> ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_3924_diff__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus @ nat @ M2 @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [K3: nat] : K3
@ ( minus_minus @ nat @ M2 @ N ) ) ) ).
% diff_Suc
thf(fact_3925_card__2__iff,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X4: A,Y3: A] :
( ( S3
= ( insert2 @ A @ X4 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X4 != Y3 ) ) ) ) ).
% card_2_iff
thf(fact_3926_card__3__iff,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X4: A,Y3: A,Z4: A] :
( ( S3
= ( insert2 @ A @ X4 @ ( insert2 @ A @ Y3 @ ( insert2 @ A @ Z4 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X4 != Y3 )
& ( Y3 != Z4 )
& ( X4 != Z4 ) ) ) ) ).
% card_3_iff
thf(fact_3927_odd__card__imp__not__empty,axiom,
! [A: $tType,A6: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A6 ) )
=> ( A6
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_3928_card_Oremove,axiom,
! [A: $tType,A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( finite_card @ A @ A6 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_3929_card_Oinsert__remove,axiom,
! [A: $tType,A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_3930_card__Suc__Diff1,axiom,
! [A: $tType,A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A6 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_3931_card__Diff1__less,axiom,
! [A: $tType,A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) ) ) ) ).
% card_Diff1_less
thf(fact_3932_card__Diff2__less,axiom,
! [A: $tType,A6: set @ A,X3: A,Y: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( member @ A @ Y @ A6 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_3933_card__Diff1__less__iff,axiom,
! [A: $tType,A6: set @ A,X3: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) )
= ( ( finite_finite2 @ A @ A6 )
& ( member @ A @ X3 @ A6 ) ) ) ).
% card_Diff1_less_iff
thf(fact_3934_bit__numeral__rec_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W ) ) @ N )
= ( case_nat @ $o @ $false @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) ) @ N ) ) ) ).
% bit_numeral_rec(1)
thf(fact_3935_card__Un__disjoint,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A6 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_3936_bit__numeral__rec_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W ) ) @ N )
= ( case_nat @ $o @ $true @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) ) @ N ) ) ) ).
% bit_numeral_rec(2)
thf(fact_3937_card__Diff__singleton,axiom,
! [A: $tType,X3: A,A6: set @ A] :
( ( member @ A @ X3 @ A6 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_3938_card__Diff__singleton__if,axiom,
! [A: $tType,X3: A,A6: set @ A] :
( ( ( member @ A @ X3 @ A6 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X3 @ A6 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A6 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_3939_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A6: set @ B,F3: B > A,N: A,K2: nat] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) )
& ( ord_less_eq @ A @ ( F3 @ I3 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A6 ) @ K2 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) @ ( power_power @ A @ N @ K2 ) ) ) ) ) ) ).
% prod_le_power
thf(fact_3940_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A6: set @ B,F3: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( divide_divide @ A @ K5 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_3941_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y: set @ A,X3: 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 @ X3 @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_3942_polyfun__roots__card,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,K2: nat,N: nat] :
( ( ( C3 @ K2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ).
% polyfun_roots_card
thf(fact_3943_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A3: B,B2: B > A,C3: A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ C3 )
@ S3 )
= ( times_times @ A @ ( B2 @ A3 ) @ ( power_power @ A @ C3 @ ( minus_minus @ nat @ ( finite_card @ B @ S3 ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ C3 )
@ S3 )
= ( power_power @ A @ C3 @ ( finite_card @ B @ S3 ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_3944_length__n__lists,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( n_lists @ A @ N @ Xs2 ) )
= ( power_power @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ).
% length_n_lists
thf(fact_3945_polyfun__rootbound,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,K2: nat,N: nat] :
( ( ( C3 @ K2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
& ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ) ).
% polyfun_rootbound
thf(fact_3946_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_3947_card__lists__distinct__length__eq,axiom,
! [A: $tType,A6: set @ A,K2: nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ nat @ K2 @ ( finite_card @ A @ A6 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= K2 )
& ( distinct @ A @ Xs )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A6 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ K2 ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A6 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_3948_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K2: nat,A6: set @ A] :
( ( ord_less @ nat @ K2 @ ( finite_card @ A @ A6 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= K2 )
& ( distinct @ A @ Xs )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A6 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ K2 ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A6 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_3949_Code__Numeral_Opositive__def,axiom,
( code_positive
= ( numeral_numeral @ code_integer ) ) ).
% Code_Numeral.positive_def
thf(fact_3950_distinct__swap,axiom,
! [A: $tType,I: nat,Xs2: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs2 @ I @ ( nth @ A @ Xs2 @ J ) ) @ J @ ( nth @ A @ Xs2 @ I ) ) )
= ( distinct @ A @ Xs2 ) ) ) ) ).
% distinct_swap
thf(fact_3951_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A6: set @ A,N: nat] :
( ( finite_finite2 @ A @ A6 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N )
& ( distinct @ A @ Xs )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A6 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_3952_finite__distinct__list,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ? [Xs3: list @ A] :
( ( ( set2 @ A @ Xs3 )
= A6 )
& ( distinct @ A @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_3953_distinct__concat,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( distinct @ ( list @ A ) @ Xs2 )
=> ( ! [Ys4: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs2 ) )
=> ( distinct @ A @ Ys4 ) )
=> ( ! [Ys4: list @ A,Zs: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs2 ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( set2 @ ( list @ A ) @ Xs2 ) )
=> ( ( Ys4 != Zs )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys4 ) @ ( set2 @ A @ Zs ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs2 ) ) ) ) ) ).
% distinct_concat
thf(fact_3954_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs2: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Xs2 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( nth @ A @ Xs2 @ I )
= ( nth @ A @ Xs2 @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_3955_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs: list @ A] :
! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( I4 != J3 )
=> ( ( nth @ A @ Xs @ I4 )
!= ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_3956_card__distinct,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( finite_card @ A @ ( set2 @ A @ Xs2 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( distinct @ A @ Xs2 ) ) ).
% card_distinct
thf(fact_3957_distinct__card,axiom,
! [A: $tType,Xs2: list @ A] :
( ( distinct @ A @ Xs2 )
=> ( ( finite_card @ A @ ( set2 @ A @ Xs2 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% distinct_card
thf(fact_3958_distinct__Ex1,axiom,
! [A: $tType,Xs2: list @ A,X3: A] :
( ( distinct @ A @ Xs2 )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ? [X5: nat] :
( ( ord_less @ nat @ X5 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ( nth @ A @ Xs2 @ X5 )
= X3 )
& ! [Y6: nat] :
( ( ( ord_less @ nat @ Y6 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ( nth @ A @ Xs2 @ Y6 )
= X3 ) )
=> ( Y6 = X5 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_3959_bij__betw__nth,axiom,
! [A: $tType,Xs2: list @ A,A6: set @ nat,B5: set @ A] :
( ( distinct @ A @ Xs2 )
=> ( ( A6
= ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
=> ( ( B5
= ( set2 @ A @ Xs2 ) )
=> ( bij_betw @ nat @ A @ ( nth @ A @ Xs2 ) @ A6 @ B5 ) ) ) ) ).
% bij_betw_nth
thf(fact_3960_distinct__list__update,axiom,
! [A: $tType,Xs2: list @ A,A3: A,I: nat] :
( ( distinct @ A @ Xs2 )
=> ( ~ ( member @ A @ A3 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( insert2 @ A @ ( nth @ A @ Xs2 @ I ) @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( distinct @ A @ ( list_update @ A @ Xs2 @ I @ A3 ) ) ) ) ).
% distinct_list_update
thf(fact_3961_set__update__distinct,axiom,
! [A: $tType,Xs2: list @ A,N: nat,X3: A] :
( ( distinct @ A @ Xs2 )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs2 @ N @ X3 ) )
= ( insert2 @ A @ X3 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( insert2 @ A @ ( nth @ A @ Xs2 @ N ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_3962_bit__cut__integer__def,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
@ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K3 ) ) ) ) ).
% bit_cut_integer_def
thf(fact_3963_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_3964_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K3 @ L2 ) @ ( modulo_modulo @ code_integer @ K3 @ L2 ) ) ) ) ).
% divmod_integer_def
thf(fact_3965_card__disjoint__shuffles,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs2 @ Ys ) )
= ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% card_disjoint_shuffles
thf(fact_3966_length__shuffles,axiom,
! [A: $tType,Zs2: list @ A,Xs2: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs2 @ Ys ) )
=> ( ( size_size @ ( list @ A ) @ Zs2 )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ) ).
% length_shuffles
thf(fact_3967_set__shuffles,axiom,
! [A: $tType,Zs2: list @ A,Xs2: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs2 @ Ys ) )
=> ( ( set2 @ A @ Zs2 )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_3968_distinct__disjoint__shuffles,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( distinct @ A @ Xs2 )
=> ( ( distinct @ A @ Ys )
=> ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs2 @ Ys ) )
=> ( distinct @ A @ Zs2 ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_3969_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,S7: 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 ) @ S7 ) )
@ ( S7
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_3970_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_3971_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_3972_nat_Osplit__sels_I2_J,axiom,
! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ~ ( ( ( Nat
= ( zero_zero @ nat ) )
& ~ ( P @ F1 ) )
| ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
& ~ ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ).
% nat.split_sels(2)
thf(fact_3973_int__of__integer__numeral,axiom,
! [K2: num] :
( ( code_int_of_integer @ ( numeral_numeral @ code_integer @ K2 ) )
= ( numeral_numeral @ int @ K2 ) ) ).
% int_of_integer_numeral
thf(fact_3974_nat__of__integer__code__post_I3_J,axiom,
! [K2: num] :
( ( code_nat_of_integer @ ( numeral_numeral @ code_integer @ K2 ) )
= ( numeral_numeral @ nat @ K2 ) ) ).
% nat_of_integer_code_post(3)
thf(fact_3975_divmod__abs__code_I5_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ J @ ( zero_zero @ code_integer ) )
= ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( abs_abs @ code_integer @ J ) ) ) ).
% divmod_abs_code(5)
thf(fact_3976_divmod__abs__code_I6_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ ( zero_zero @ code_integer ) @ J )
= ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ) ).
% divmod_abs_code(6)
thf(fact_3977_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K3: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L2 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L2 ) ) ) ) ) ).
% divmod_abs_def
thf(fact_3978_pred__def,axiom,
( pred
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [X23: nat] : X23 ) ) ).
% pred_def
thf(fact_3979_nat_Osplit__sels_I1_J,axiom,
! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ( ( Nat
= ( zero_zero @ nat ) )
=> ( P @ F1 ) )
& ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
=> ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ).
% nat.split_sels(1)
thf(fact_3980_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,S7: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S7
= ( 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 @ S7 ) ) )
@ ( 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,S7: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S7
= ( 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 ) @ S7 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_3981_card__Pow,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_card @ ( set @ A ) @ ( pow2 @ A @ A6 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A6 ) ) ) ) ).
% card_Pow
thf(fact_3982_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_3983_bezw__0,axiom,
! [X3: nat] :
( ( bezw @ X3 @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_3984_Pow__singleton__iff,axiom,
! [A: $tType,X6: set @ A,Y8: set @ A] :
( ( ( pow2 @ A @ X6 )
= ( insert2 @ ( set @ A ) @ Y8 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X6
= ( bot_bot @ ( set @ A ) ) )
& ( Y8
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_3985_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_3986_PowI,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( member @ ( set @ A ) @ A6 @ ( pow2 @ A @ B5 ) ) ) ).
% PowI
thf(fact_3987_Pow__iff,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( member @ ( set @ A ) @ A6 @ ( pow2 @ A @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A6 @ B5 ) ) ).
% Pow_iff
thf(fact_3988_Pow__Int__eq,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( pow2 @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) )
= ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A6 ) @ ( pow2 @ A @ B5 ) ) ) ).
% Pow_Int_eq
thf(fact_3989_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > B,X3: A,Y: C] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_Pair @ A @ C @ X3 @ Y ) )
= ( product_Pair @ A @ B @ X3 @ ( F3 @ Y ) ) ) ).
% apsnd_conv
thf(fact_3990_old_Onat_Osimps_I7_J,axiom,
! [T: $tType,F1: T,F22: nat > T > T,Nat: nat] :
( ( rec_nat @ T @ F1 @ F22 @ ( suc @ Nat ) )
= ( F22 @ Nat @ ( rec_nat @ T @ F1 @ F22 @ Nat ) ) ) ).
% old.nat.simps(7)
thf(fact_3991_old_Onat_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: nat > T > T] :
( ( rec_nat @ T @ F1 @ F22 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(6)
thf(fact_3992_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_3993_Pow__top,axiom,
! [A: $tType,A6: set @ A] : ( member @ ( set @ A ) @ A6 @ ( pow2 @ A @ A6 ) ) ).
% Pow_top
thf(fact_3994_Pow__not__empty,axiom,
! [A: $tType,A6: set @ A] :
( ( pow2 @ A @ A6 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_3995_Pow__bottom,axiom,
! [A: $tType,B5: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow2 @ A @ B5 ) ) ).
% Pow_bottom
thf(fact_3996_PowD,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( member @ ( set @ A ) @ A6 @ ( pow2 @ A @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B5 ) ) ).
% PowD
thf(fact_3997_Pow__mono,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A6 ) @ ( pow2 @ A @ B5 ) ) ) ).
% Pow_mono
thf(fact_3998_Un__Pow__subset,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A6 ) @ ( pow2 @ A @ B5 ) ) @ ( pow2 @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) ) ) ).
% Un_Pow_subset
thf(fact_3999_Pow__def,axiom,
! [A: $tType] :
( ( pow2 @ A )
= ( ^ [A7: set @ A] :
( collect @ ( set @ A )
@ ^ [B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% Pow_def
thf(fact_4000_old_Orec__nat__def,axiom,
! [T: $tType] :
( ( rec_nat @ T )
= ( ^ [F12: T,F23: nat > T > T,X4: nat] : ( the @ T @ ( rec_set_nat @ T @ F12 @ F23 @ X4 ) ) ) ) ).
% old.rec_nat_def
thf(fact_4001_rec__nat__Suc__imp,axiom,
! [A: $tType,F3: nat > A,F1: A,F22: nat > A > A,N: nat] :
( ( F3
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F3 @ ( suc @ N ) )
= ( F22 @ N @ ( F3 @ N ) ) ) ) ).
% rec_nat_Suc_imp
thf(fact_4002_card__partition,axiom,
! [A: $tType,C4: set @ ( set @ A ),K2: nat] :
( ( finite_finite2 @ ( set @ A ) @ C4 )
=> ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) )
=> ( ! [C2: set @ A] :
( ( member @ ( set @ A ) @ C2 @ C4 )
=> ( ( finite_card @ A @ C2 )
= K2 ) )
=> ( ! [C1: set @ A,C22: set @ A] :
( ( member @ ( set @ A ) @ C1 @ C4 )
=> ( ( member @ ( set @ A ) @ C22 @ C4 )
=> ( ( C1 != C22 )
=> ( ( inf_inf @ ( set @ A ) @ C1 @ C22 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( times_times @ nat @ K2 @ ( finite_card @ ( set @ A ) @ C4 ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_4003_drop__bit__numeral__minus__bit1,axiom,
! [L: num,K2: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K2 ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K2 ) ) ) ) ) ).
% drop_bit_numeral_minus_bit1
thf(fact_4004_drop__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
= ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ A3 ) ) ) ).
% drop_bit_drop_bit
thf(fact_4005_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= Y ) ) ) ).
% Sup_atLeastAtMost
thf(fact_4006_drop__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K2: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K2 ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( numeral_numeral @ A @ K2 ) ) ) ) ).
% drop_bit_Suc_bit0
thf(fact_4007_drop__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K2: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K2 ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( numeral_numeral @ A @ K2 ) ) ) ) ).
% drop_bit_Suc_bit1
thf(fact_4008_drop__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K2: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K2 ) ) )
= ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K2 ) ) ) ) ).
% drop_bit_numeral_bit0
thf(fact_4009_drop__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K2: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K2 ) ) )
= ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K2 ) ) ) ) ).
% drop_bit_numeral_bit1
thf(fact_4010_drop__bit__Suc__minus__bit0,axiom,
! [N: nat,K2: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K2 ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) ) ) ).
% drop_bit_Suc_minus_bit0
thf(fact_4011_drop__bit__numeral__minus__bit0,axiom,
! [L: num,K2: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K2 ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) ) ) ).
% drop_bit_numeral_minus_bit0
thf(fact_4012_drop__bit__Suc__minus__bit1,axiom,
! [N: nat,K2: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K2 ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K2 ) ) ) ) ) ).
% drop_bit_Suc_minus_bit1
thf(fact_4013_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B5: set @ A,A3: A] :
( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B5 ) @ A3 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B5 )
=> ( ( inf_inf @ A @ X4 @ A3 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_4014_insert__partition,axiom,
! [A: $tType,X3: set @ A,F6: set @ ( set @ A )] :
( ~ ( member @ ( set @ A ) @ X3 @ F6 )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ ( insert2 @ ( set @ A ) @ X3 @ F6 ) )
=> ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ ( insert2 @ ( set @ A ) @ X3 @ F6 ) )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X5 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ ( complete_Sup_Sup @ ( set @ A ) @ F6 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_partition
thf(fact_4015_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_4016_take__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ A3 ) ) ) ) ).
% take_bit_drop_bit
thf(fact_4017_finite__Sup__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ A6 )
=> ( ( member @ A @ Y4 @ A6 )
=> ( member @ A @ ( sup_sup @ A @ X5 @ Y4 ) @ A6 ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A6 ) @ A6 ) ) ) ) ) ).
% finite_Sup_in
thf(fact_4018_card__Union__le__sum__card__weak,axiom,
! [A: $tType,U3: set @ ( set @ A )] :
( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ U3 )
=> ( finite_finite2 @ A @ X5 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U3 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U3 ) ) ) ).
% card_Union_le_sum_card_weak
thf(fact_4019_finite__subset__Union,axiom,
! [A: $tType,A6: set @ A,B11: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) )
=> ~ ! [F7: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F7 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F7 @ B11 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ F7 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_4020_bits__ident,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= A3 ) ) ).
% bits_ident
thf(fact_4021_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,L: A,E3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X5 @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S3 ) @ L ) ) @ E3 ) ) ) ) ).
% cSup_asclose
thf(fact_4022_stable__imp__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( bit_se4197421643247451524op_bit @ A @ N @ A3 )
= A3 ) ) ) ).
% stable_imp_drop_bit_eq
thf(fact_4023_drop__bit__half,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( divide_divide @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% drop_bit_half
thf(fact_4024_subset__CollectI,axiom,
! [A: $tType,B5: set @ A,A6: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B5 )
=> ( ( Q @ X5 )
=> ( P @ X5 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ B5 )
& ( Q @ X4 ) ) )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_4025_subset__Collect__iff,axiom,
! [A: $tType,B5: set @ A,A6: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P @ X4 ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B5 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_4026_dvd__partition,axiom,
! [A: $tType,C4: set @ ( set @ A ),K2: nat] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C4 )
=> ( dvd_dvd @ nat @ K2 @ ( finite_card @ A @ X5 ) ) )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C4 )
=> ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ C4 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X5 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( dvd_dvd @ nat @ K2 @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_4027_drop__bit__int__def,axiom,
( ( bit_se4197421643247451524op_bit @ int )
= ( ^ [N3: nat,K3: int] : ( divide_divide @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% drop_bit_int_def
thf(fact_4028_drop__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ A3 )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% drop_bit_Suc
thf(fact_4029_drop__bit__eq__div,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N3: nat,A8: A] : ( divide_divide @ A @ A8 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% drop_bit_eq_div
thf(fact_4030_even__drop__bit__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% even_drop_bit_iff_not_bit
thf(fact_4031_bit__iff__odd__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A8: A,N3: nat] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A8 ) ) ) ) ) ).
% bit_iff_odd_drop_bit
thf(fact_4032_slice__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,M2: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) )
= ( bit_se5824344872417868541ns_and @ A @ A3 @ ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ M2 @ N ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ) ).
% slice_eq_mask
thf(fact_4033_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N3: nat,A8: A] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ A8
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_4034_Sup__insert,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: A,A6: set @ A] :
( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ A6 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ).
% Sup_insert
thf(fact_4035_cSup__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A] :
( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% cSup_singleton
thf(fact_4036_ccpo__Sup__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X3: A] :
( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% ccpo_Sup_singleton
thf(fact_4037_cSup__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X3 ) )
= X3 ) ) ) ).
% cSup_atLeastAtMost
thf(fact_4038_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ A6 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( X4
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_4039_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A] :
( ( ( complete_Sup_Sup @ A @ A6 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( X4
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_4040_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_4041_Sup__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Sup_empty
thf(fact_4042_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_4043_drop__bit__nat__def,axiom,
( ( bit_se4197421643247451524op_bit @ nat )
= ( ^ [N3: nat,M5: nat] : ( divide_divide @ nat @ M5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% drop_bit_nat_def
thf(fact_4044_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B2 )
=> ? [C2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
& ( ord_less_eq @ A @ C2 @ B2 )
& ! [X: A] :
( ( ( ord_less_eq @ A @ A3 @ X )
& ( ord_less @ A @ X @ C2 ) )
=> ( P @ X ) )
& ! [D6: A] :
( ! [X5: A] :
( ( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less @ A @ X5 @ D6 ) )
=> ( P @ X5 ) )
=> ( ord_less_eq @ A @ D6 @ C2 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_4045_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_bot @ A ) )
=> ! [X6: set @ A,A3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ( ord_less_eq @ A @ X5 @ A3 ) )
=> ( ! [Y4: A] :
( ! [X: A] :
( ( member @ A @ X @ X6 )
=> ( ord_less_eq @ A @ X @ Y4 ) )
=> ( ord_less_eq @ A @ A3 @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= A3 ) ) ) ) ).
% cSup_eq
thf(fact_4046_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z2: A,X6: set @ A] :
( ( member @ A @ Z2 @ X6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ( ord_less_eq @ A @ X5 @ Z2 ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= Z2 ) ) ) ) ).
% cSup_eq_maximum
thf(fact_4047_Sup__upper2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A6: set @ A,V2: A] :
( ( member @ A @ U @ A6 )
=> ( ( ord_less_eq @ A @ V2 @ U )
=> ( ord_less_eq @ A @ V2 @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% Sup_upper2
thf(fact_4048_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,B2: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ B2 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ X4 @ B2 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_4049_Sup__upper,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,A6: set @ A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ).
% Sup_upper
thf(fact_4050_Sup__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,Z2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ A @ X5 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ Z2 ) ) ) ).
% Sup_least
thf(fact_4051_Sup__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ? [X: A] :
( ( member @ A @ X @ B5 )
& ( ord_less_eq @ A @ A5 @ X ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ).
% Sup_mono
thf(fact_4052_Sup__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,X3: A] :
( ! [Y4: A] :
( ( member @ A @ Y4 @ A6 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ! [Y4: A] :
( ! [Z5: A] :
( ( member @ A @ Z5 @ A6 )
=> ( ord_less_eq @ A @ Z5 @ Y4 ) )
=> ( ord_less_eq @ A @ X3 @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ A6 )
= X3 ) ) ) ) ).
% Sup_eqI
thf(fact_4053_empty__Union__conv,axiom,
! [A: $tType,A6: set @ ( set @ A )] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ A6 ) )
= ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A6 )
=> ( X4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% empty_Union_conv
thf(fact_4054_Union__empty__conv,axiom,
! [A: $tType,A6: set @ ( set @ A )] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ A6 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A6 )
=> ( X4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_empty_conv
thf(fact_4055_Union__subsetI,axiom,
! [A: $tType,A6: set @ ( set @ A ),B5: set @ ( set @ A )] :
( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A6 )
=> ? [Y6: set @ A] :
( ( member @ ( set @ A ) @ Y6 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ X5 @ Y6 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A6 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B5 ) ) ) ).
% Union_subsetI
thf(fact_4056_Union__upper,axiom,
! [A: $tType,B5: set @ A,A6: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B5 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ ( complete_Sup_Sup @ ( set @ A ) @ A6 ) ) ) ).
% Union_upper
thf(fact_4057_Union__least,axiom,
! [A: $tType,A6: set @ ( set @ A ),C4: set @ A] :
( ! [X10: set @ A] :
( ( member @ ( set @ A ) @ X10 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ X10 @ C4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A6 ) @ C4 ) ) ).
% Union_least
thf(fact_4058_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X3: A,A6: set @ A] :
( ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A6 ) )
= ( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X3 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ord_less @ A @ Y3 @ X4 ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_4059_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,U: A] :
( ! [V: A] :
( ( member @ A @ V @ A6 )
=> ( ord_less_eq @ A @ U @ V ) )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_4060_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,Z2: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ( ord_less_eq @ A @ X5 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X6 ) @ Z2 ) ) ) ) ).
% cSup_least
thf(fact_4061_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,A3: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ( ord_less_eq @ A @ X5 @ A3 ) )
=> ( ! [Y4: A] :
( ! [X: A] :
( ( member @ A @ X @ X6 )
=> ( ord_less_eq @ A @ X @ Y4 ) )
=> ( ord_less_eq @ A @ A3 @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= A3 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_4062_le__cSup__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,X3: A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ X6 ) ) ) ) ) ).
% le_cSup_finite
thf(fact_4063_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ).
% Sup_subset_mono
thf(fact_4064_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,Z2: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z2 @ ( complete_Sup_Sup @ A @ X6 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ X6 )
& ( ord_less @ A @ Z2 @ X5 ) ) ) ) ) ).
% less_cSupD
thf(fact_4065_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y: A,X6: set @ A] :
( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X6 ) )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ~ ( ord_less @ A @ Y @ X5 ) ) ) ) ) ).
% less_cSupE
thf(fact_4066_Union__disjoint,axiom,
! [A: $tType,C4: set @ ( set @ A ),A6: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) @ A6 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ X4 @ A6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_disjoint
thf(fact_4067_Union__mono,axiom,
! [A: $tType,A6: set @ ( set @ A ),B5: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A6 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A6 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B5 ) ) ) ).
% Union_mono
thf(fact_4068_Union__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Union_empty
thf(fact_4069_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,A3: A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X6 ) @ A3 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ X6 )
=> ( ord_less @ A @ X4 @ A3 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_4070_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,A3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X5 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S3 ) ) @ A3 ) ) ) ) ).
% cSup_abs_le
thf(fact_4071_Sup__inter__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,B5: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ).
% Sup_inter_less_eq
thf(fact_4072_Union__Int__subset,axiom,
! [A: $tType,A6: set @ ( set @ A ),B5: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A6 @ B5 ) ) @ ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A6 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B5 ) ) ) ).
% Union_Int_subset
thf(fact_4073_Sup__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S3: set @ A,X3: A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( ( S3
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X3 @ S3 ) )
= X3 ) )
& ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X3 @ S3 ) )
= ( ord_max @ A @ X3 @ ( complete_Sup_Sup @ A @ S3 ) ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_4074_ccSup__empty,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% ccSup_empty
thf(fact_4075_card__UNION,axiom,
! [A: $tType,A6: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A6 )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A6 )
=> ( finite_finite2 @ A @ X5 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A6 ) )
= ( 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 @ A6 )
& ( I7
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_4076_Sup__finite__insert,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ! [A3: A,A6: set @ A] :
( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ A6 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ).
% Sup_finite_insert
thf(fact_4077_Suc__0__mod__numeral,axiom,
! [K2: num] :
( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K2 ) )
= ( product_snd @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K2 ) ) ) ).
% Suc_0_mod_numeral
thf(fact_4078_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,X3: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F3 @ X3 ) )
= ( F3 @ ( product_snd @ B @ C @ X3 ) ) ) ).
% snd_apsnd
thf(fact_4079_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,X3: product_prod @ A @ C,G3: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F3 @ X3 )
= ( product_apsnd @ C @ B @ A @ G3 @ X3 ) )
= ( ( F3 @ ( product_snd @ A @ C @ X3 ) )
= ( G3 @ ( product_snd @ A @ C @ X3 ) ) ) ) ).
% apsnd_eq_conv
thf(fact_4080_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A6: set @ A] :
( ( ( complete_Inf_Inf @ A @ A6 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X4 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A6 )
& ( ord_less @ A @ Y3 @ X4 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_4081_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= X3 ) ) ) ).
% Inf_atLeastAtMost
thf(fact_4082_cInf__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X3 ) )
= Y ) ) ) ).
% cInf_atLeastAtMost
thf(fact_4083_cInf__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A] :
( ( complete_Inf_Inf @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% cInf_singleton
thf(fact_4084_Inf__insert,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: A,A6: set @ A] :
( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ A6 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ A6 ) ) ) ) ).
% Inf_insert
thf(fact_4085_Inf__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atMost @ A @ X3 ) )
= ( bot_bot @ A ) ) ) ).
% Inf_atMost
thf(fact_4086_numeral__mod__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [K2: num,L: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ K2 ) @ ( numeral_numeral @ A @ L ) )
= ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ K2 @ L ) ) ) ) ).
% numeral_mod_numeral
thf(fact_4087_one__mod__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% one_mod_numeral
thf(fact_4088_Inf__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,X3: A] :
( ! [I3: A] :
( ( member @ A @ I3 @ A6 )
=> ( ord_less_eq @ A @ X3 @ I3 ) )
=> ( ! [Y4: A] :
( ! [I2: A] :
( ( member @ A @ I2 @ A6 )
=> ( ord_less_eq @ A @ Y4 @ I2 ) )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ( complete_Inf_Inf @ A @ A6 )
= X3 ) ) ) ) ).
% Inf_eqI
thf(fact_4089_Inf__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: set @ A,A6: set @ A] :
( ! [B4: A] :
( ( member @ A @ B4 @ B5 )
=> ? [X: A] :
( ( member @ A @ X @ A6 )
& ( ord_less_eq @ A @ X @ B4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B5 ) ) ) ) ).
% Inf_mono
thf(fact_4090_Inf__lower,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,A6: set @ A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ X3 ) ) ) ).
% Inf_lower
thf(fact_4091_Inf__lower2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A6: set @ A,V2: A] :
( ( member @ A @ U @ A6 )
=> ( ( ord_less_eq @ A @ U @ V2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ V2 ) ) ) ) ).
% Inf_lower2
thf(fact_4092_le__Inf__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: A,A6: set @ A] :
( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A6 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ B2 @ X4 ) ) ) ) ) ).
% le_Inf_iff
thf(fact_4093_Inf__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,Z2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ A @ Z2 @ X5 ) )
=> ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ A6 ) ) ) ) ).
% Inf_greatest
thf(fact_4094_cInf__eq__minimum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z2: A,X6: set @ A] :
( ( member @ A @ Z2 @ X6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ( ord_less_eq @ A @ Z2 @ X5 ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= Z2 ) ) ) ) ).
% cInf_eq_minimum
thf(fact_4095_cInf__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_top @ A ) )
=> ! [X6: set @ A,A3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ( ord_less_eq @ A @ A3 @ X5 ) )
=> ( ! [Y4: A] :
( ! [X: A] :
( ( member @ A @ X @ X6 )
=> ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ord_less_eq @ A @ Y4 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= A3 ) ) ) ) ).
% cInf_eq
thf(fact_4096_Inter__lower,axiom,
! [A: $tType,B5: set @ A,A6: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B5 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A6 ) @ B5 ) ) ).
% Inter_lower
thf(fact_4097_Inter__greatest,axiom,
! [A: $tType,A6: set @ ( set @ A ),C4: set @ A] :
( ! [X10: set @ A] :
( ( member @ ( set @ A ) @ X10 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ C4 @ X10 ) )
=> ( ord_less_eq @ ( set @ A ) @ C4 @ ( complete_Inf_Inf @ ( set @ A ) @ A6 ) ) ) ).
% Inter_greatest
thf(fact_4098_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_4099_snd__eqD,axiom,
! [B: $tType,A: $tType,X3: B,Y: A,A3: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) )
= A3 )
=> ( Y = A3 ) ) ).
% snd_eqD
thf(fact_4100_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X15: A,X23: B] : X23 ) ) ).
% snd_def
thf(fact_4101_Inf__finite__insert,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ! [A3: A,A6: set @ A] :
( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ A6 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ A6 ) ) ) ) ).
% Inf_finite_insert
thf(fact_4102_Inf__le__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A6: set @ A,X3: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ X3 )
= ( ! [Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ord_less @ A @ X4 @ Y3 ) ) ) ) ) ) ).
% Inf_le_iff
thf(fact_4103_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,U: A] :
( ! [V: A] :
( ( member @ A @ V @ A6 )
=> ( ord_less_eq @ A @ V @ U ) )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_4104_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,Z2: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ( ord_less_eq @ A @ Z2 @ X5 ) )
=> ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ X6 ) ) ) ) ) ).
% cInf_greatest
thf(fact_4105_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,A3: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ( ord_less_eq @ A @ A3 @ X5 ) )
=> ( ! [Y4: A] :
( ! [X: A] :
( ( member @ A @ X @ X6 )
=> ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ord_less_eq @ A @ Y4 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= A3 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_4106_cInf__le__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,X3: A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X6 ) @ X3 ) ) ) ) ).
% cInf_le_finite
thf(fact_4107_Inf__superset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: set @ A,A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B5 ) ) ) ) ).
% Inf_superset_mono
thf(fact_4108_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,Z2: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X6 ) @ Z2 )
=> ? [X5: A] :
( ( member @ A @ X5 @ X6 )
& ( ord_less @ A @ X5 @ Z2 ) ) ) ) ) ).
% cInf_lessD
thf(fact_4109_Inter__anti__mono,axiom,
! [A: $tType,B5: set @ ( set @ A ),A6: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ B5 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A6 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B5 ) ) ) ).
% Inter_anti_mono
thf(fact_4110_Inter__subset,axiom,
! [A: $tType,A6: set @ ( set @ A ),B5: set @ A] :
( ! [X10: set @ A] :
( ( member @ ( set @ A ) @ X10 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ X10 @ B5 ) )
=> ( ( A6
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A6 ) @ B5 ) ) ) ).
% Inter_subset
thf(fact_4111_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,A3: A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X6 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ X6 )
=> ( ord_less @ A @ A3 @ X4 ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_4112_Inf__le__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ).
% Inf_le_Sup
thf(fact_4113_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,A3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X5 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S3 ) ) @ A3 ) ) ) ) ).
% cInf_abs_ge
thf(fact_4114_less__eq__Inf__inter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,B5: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B5 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ) ) ).
% less_eq_Inf_inter
thf(fact_4115_finite__Inf__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ A6 )
=> ( ( member @ A @ Y4 @ A6 )
=> ( member @ A @ ( inf_inf @ A @ X5 @ Y4 ) @ A6 ) ) )
=> ( member @ A @ ( complete_Inf_Inf @ A @ A6 ) @ A6 ) ) ) ) ) ).
% finite_Inf_in
thf(fact_4116_snd__divmod,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ M2 @ N ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% snd_divmod
thf(fact_4117_Inter__Un__subset,axiom,
! [A: $tType,A6: set @ ( set @ A ),B5: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A6 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B5 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A6 @ B5 ) ) ) ).
% Inter_Un_subset
thf(fact_4118_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,L: A,E3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X5 @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S3 ) @ L ) ) @ E3 ) ) ) ) ).
% cInf_asclose
thf(fact_4119_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F4: A > nat,R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y3: A] :
( ( ord_less @ nat @ ( F4 @ X4 ) @ ( F4 @ Y3 ) )
| ( ( ord_less_eq @ nat @ ( F4 @ X4 ) @ ( F4 @ Y3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R6 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_4120_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K3: nat,M5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M5 @ K3 ) @ ( product_Pair @ nat @ nat @ M5 @ ( minus_minus @ nat @ K3 @ M5 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus @ nat @ M5 @ ( suc @ K3 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_4121_prod__decode__aux_Oelims,axiom,
! [X3: nat,Xa2: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
= Y )
=> ( ( ( ord_less_eq @ nat @ Xa2 @ X3 )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa2 @ ( minus_minus @ nat @ X3 @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa2 @ X3 )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus @ nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_4122_Suc__0__div__numeral,axiom,
! [K2: num] :
( ( divide_divide @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K2 ) )
= ( product_fst @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K2 ) ) ) ).
% Suc_0_div_numeral
thf(fact_4123_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,X3: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F3 @ X3 ) )
= ( product_fst @ A @ C @ X3 ) ) ).
% fst_apsnd
thf(fact_4124_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_4125_numeral__div__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [K2: num,L: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ K2 ) @ ( numeral_numeral @ A @ L ) )
= ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ K2 @ L ) ) ) ) ).
% numeral_div_numeral
thf(fact_4126_one__div__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% one_div_numeral
thf(fact_4127_Inf__nat__def1,axiom,
! [K5: set @ nat] :
( ( K5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( complete_Inf_Inf @ nat @ K5 ) @ K5 ) ) ).
% Inf_nat_def1
thf(fact_4128_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_4129_prod__eqI,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P2 )
= ( product_fst @ A @ B @ Q3 ) )
=> ( ( ( product_snd @ A @ B @ P2 )
= ( product_snd @ A @ B @ Q3 ) )
=> ( P2 = Q3 ) ) ) ).
% prod_eqI
thf(fact_4130_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y5: product_prod @ A @ B,Z: product_prod @ A @ B] : Y5 = Z )
= ( ^ [S7: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S7 )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S7 )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_4131_fst__eqD,axiom,
! [B: $tType,A: $tType,X3: A,Y: B,A3: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y ) )
= A3 )
=> ( X3 = A3 ) ) ).
% fst_eqD
thf(fact_4132_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X2: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_4133_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X15: A,X23: B] : X15 ) ) ).
% fst_def
thf(fact_4134_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X3: A,Y: B,A3: product_prod @ A @ B] :
( ( P @ X3 @ Y )
=> ( ( A3
= ( product_Pair @ A @ B @ X3 @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_4135_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( T2
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% surjective_pairing
thf(fact_4136_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_4137_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F4: B > C > A,P5: product_prod @ B @ C] : ( F4 @ ( product_fst @ B @ C @ P5 ) @ ( product_snd @ B @ C @ P5 ) ) ) ) ).
% case_prod_beta
thf(fact_4138_split__beta,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,Prod3: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_4139_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X3: product_prod @ A @ B,A6: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) )
=> ( A6 @ ( product_fst @ A @ B @ X3 ) @ ( product_snd @ A @ B @ X3 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_4140_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C6: A > B > C,P5: product_prod @ A @ B] : ( C6 @ ( product_fst @ A @ B @ P5 ) @ ( product_snd @ A @ B @ P5 ) ) ) ) ).
% case_prod_unfold
thf(fact_4141_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,X4: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ X4 ) @ ( product_snd @ A @ B @ X4 ) ) ) ) ).
% case_prod_beta'
thf(fact_4142_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F3: A > B > C,G3: D > A] :
( ( ^ [U2: product_prod @ D @ B] : ( F3 @ ( G3 @ ( product_fst @ D @ B @ U2 ) ) @ ( product_snd @ D @ B @ U2 ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X4: D] : ( F3 @ ( G3 @ X4 ) ) ) ) ).
% split_comp_eq
thf(fact_4143_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P @ ( F3 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_4144_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P @ ( F3 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_4145_fst__divmod,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ M2 @ N ) )
= ( divide_divide @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% fst_divmod
thf(fact_4146_The__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
= ( the @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% The_case_prod
thf(fact_4147_minus__one__mod__numeral,axiom,
! [N: num] :
( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) )
= ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% minus_one_mod_numeral
thf(fact_4148_one__mod__minus__numeral,axiom,
! [N: num] :
( ( modulo_modulo @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ) ).
% one_mod_minus_numeral
thf(fact_4149_minus__numeral__mod__numeral,axiom,
! [M2: num,N: num] :
( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ M2 @ N ) ) ) ) ).
% minus_numeral_mod_numeral
thf(fact_4150_numeral__mod__minus__numeral,axiom,
! [M2: num,N: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ M2 @ N ) ) ) ) ) ).
% numeral_mod_minus_numeral
thf(fact_4151_bezw__non__0,axiom,
! [Y: nat,X3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y )
=> ( ( bezw @ X3 @ Y )
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_4152_bezw_Oelims,axiom,
! [X3: nat,Xa2: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X3 @ Xa2 )
= Y )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Xa2 ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_4153_bezw_Osimps,axiom,
( bezw
= ( ^ [X4: 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 @ X4 @ Y3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X4 @ Y3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X4 @ Y3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X4 @ Y3 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_4154_in__set__enumerate__eq,axiom,
! [A: $tType,P2: product_prod @ nat @ A,N: nat,Xs2: list @ A] :
( ( member @ ( product_prod @ nat @ A ) @ P2 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs2 ) ) )
= ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P2 ) )
& ( ord_less @ nat @ ( product_fst @ nat @ A @ P2 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) )
& ( ( nth @ A @ Xs2 @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P2 ) @ N ) )
= ( product_snd @ nat @ A @ P2 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_4155_mlex__leq,axiom,
! [A: $tType,F3: A > nat,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( mlex_prod @ A @ F3 @ R ) ) ) ) ).
% mlex_leq
thf(fact_4156_mlex__iff,axiom,
! [A: $tType,X3: A,Y: A,F3: A > nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( mlex_prod @ A @ F3 @ R ) )
= ( ( ord_less @ nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X3 )
= ( F3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R ) ) ) ) ).
% mlex_iff
thf(fact_4157_mlex__less,axiom,
! [A: $tType,F3: A > nat,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( mlex_prod @ A @ F3 @ R ) ) ) ).
% mlex_less
thf(fact_4158_length__enumerate,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( size_size @ ( list @ ( product_prod @ nat @ A ) ) @ ( enumerate @ A @ N @ Xs2 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_enumerate
thf(fact_4159_nth__enumerate__eq,axiom,
! [A: $tType,M2: nat,Xs2: list @ A,N: nat] :
( ( ord_less @ nat @ M2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs2 ) @ M2 )
= ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M2 ) @ ( nth @ A @ Xs2 @ M2 ) ) ) ) ).
% nth_enumerate_eq
thf(fact_4160_in__measure,axiom,
! [A: $tType,X3: A,Y: A,F3: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measure @ A @ F3 ) )
= ( ord_less @ nat @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ).
% in_measure
thf(fact_4161_bezw_Opelims,axiom,
! [X3: nat,Xa2: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) )
=> ~ ( ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Xa2 ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) ) ) ) ).
% bezw.pelims
thf(fact_4162_in__finite__psubset,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A6 @ B5 ) @ ( finite_psubset @ A ) )
= ( ( ord_less @ ( set @ A ) @ A6 @ B5 )
& ( finite_finite2 @ A @ B5 ) ) ) ).
% in_finite_psubset
thf(fact_4163_exI__realizer,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Y: A,X3: B] :
( ( P @ Y @ X3 )
=> ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_4164_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q3: B] :
( ( P @ P2 )
=> ( ( Q @ Q3 )
=> ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q3 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q3 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_4165_prod__decode__aux_Opelims,axiom,
! [X3: nat,Xa2: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa2 @ X3 )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa2 @ ( minus_minus @ nat @ X3 @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa2 @ X3 )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus @ nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_4166_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F4: A > nat,G4: B > nat,P5: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F4 @ ( product_fst @ A @ B @ P5 ) ) @ ( G4 @ ( product_snd @ A @ B @ P5 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_4167_vebt__maxt_Opelims,axiom,
! [X3: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_maxt @ X3 )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ X3 )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A5
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A5 @ B4 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( some @ nat @ Ma2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_maxt.pelims
thf(fact_4168_vebt__mint_Opelims,axiom,
! [X3: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_mint @ X3 )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ X3 )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( ( A5
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A5 @ B4 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( some @ nat @ Mi2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_mint.pelims
thf(fact_4169_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( P @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ K @ I2 )
=> ( P @ I2 ) )
=> ( P @ K ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_4170_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q3: product_prod @ A @ B,F3: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
( ! [X5: A,Y4: B] :
( ( ( product_Pair @ A @ B @ X5 @ Y4 )
= Q3 )
=> ( ( F3 @ X5 @ Y4 )
= ( G3 @ X5 @ Y4 ) ) )
=> ( ( P2 = Q3 )
=> ( ( product_case_prod @ A @ B @ C @ F3 @ P2 )
= ( product_case_prod @ A @ B @ C @ G3 @ Q3 ) ) ) ) ).
% split_cong
thf(fact_4171_VEBT__internal_OminNull_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X3 )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
=> ( ( ( X3
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv: $o] :
( ( X3
= ( vEBT_Leaf @ $true @ Uv ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv ) ) ) )
=> ( ! [Uu: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ $true ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu @ $true ) ) ) )
=> ( ! [Uw: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) )
=> ( Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(1)
thf(fact_4172_VEBT__internal_OminNull_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
=> ( ( ( X3
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Uw: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(2)
thf(fact_4173_VEBT__internal_OminNull_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
=> ( ! [Uv: $o] :
( ( X3
= ( vEBT_Leaf @ $true @ Uv ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv ) ) )
=> ( ! [Uu: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ $true ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu @ $true ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(3)
thf(fact_4174_set__remove1__eq,axiom,
! [A: $tType,Xs2: list @ A,X3: A] :
( ( distinct @ A @ Xs2 )
=> ( ( set2 @ A @ ( remove1 @ A @ X3 @ Xs2 ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_remove1_eq
thf(fact_4175_in__set__remove1,axiom,
! [A: $tType,A3: A,B2: A,Xs2: list @ A] :
( ( A3 != B2 )
=> ( ( member @ A @ A3 @ ( set2 @ A @ ( remove1 @ A @ B2 @ Xs2 ) ) )
= ( member @ A @ A3 @ ( set2 @ A @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_4176_notin__set__remove1,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Y: A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ~ ( member @ A @ X3 @ ( set2 @ A @ ( remove1 @ A @ Y @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_4177_remove1__idem,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( remove1 @ A @ X3 @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_4178_set__remove1__subset,axiom,
! [A: $tType,X3: A,Xs2: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( remove1 @ A @ X3 @ Xs2 ) ) @ ( set2 @ A @ Xs2 ) ) ).
% set_remove1_subset
thf(fact_4179_length__remove1,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X3 @ Xs2 ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X3 @ Xs2 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% length_remove1
thf(fact_4180_nth__rotate1,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( rotate1 @ A @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( modulo_modulo @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ).
% nth_rotate1
thf(fact_4181_xor__minus__numerals_I2_J,axiom,
! [K2: int,N: num] :
( ( bit_se5824344971392196577ns_xor @ int @ K2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344971392196577ns_xor @ int @ K2 @ ( neg_numeral_sub @ int @ N @ one2 ) ) ) ) ).
% xor_minus_numerals(2)
thf(fact_4182_xor__minus__numerals_I1_J,axiom,
! [N: num,K2: int] :
( ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ K2 )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344971392196577ns_xor @ int @ ( neg_numeral_sub @ int @ N @ one2 ) @ K2 ) ) ) ).
% xor_minus_numerals(1)
thf(fact_4183_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A8: A,Xs: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( compow @ ( A > A ) @ N3 @ ( times_times @ A @ A8 ) @ ( F4 @ ( nth @ B @ Xs @ N3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_4184_Suc__funpow,axiom,
! [N: nat] :
( ( compow @ ( nat > nat ) @ N @ suc )
= ( plus_plus @ nat @ N ) ) ).
% Suc_funpow
thf(fact_4185_funpow__0,axiom,
! [A: $tType,F3: A > A,X3: A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F3 @ X3 )
= X3 ) ).
% funpow_0
thf(fact_4186_set__rotate1,axiom,
! [A: $tType,Xs2: list @ A] :
( ( set2 @ A @ ( rotate1 @ A @ Xs2 ) )
= ( set2 @ A @ Xs2 ) ) ).
% set_rotate1
thf(fact_4187_length__rotate1,axiom,
! [A: $tType,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs2 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_rotate1
thf(fact_4188_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_4189_diff__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ M2 @ N ) ) ) ).
% diff_numeral_simps(1)
thf(fact_4190_sub__num__simps_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K2 ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K2 @ L ) ) ) ) ).
% sub_num_simps(6)
thf(fact_4191_add__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ N @ M2 ) ) ) ).
% add_neg_numeral_simps(2)
thf(fact_4192_add__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ M2 @ N ) ) ) ).
% add_neg_numeral_simps(1)
thf(fact_4193_semiring__norm_I167_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Y ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ W @ V2 ) @ Y ) ) ) ).
% semiring_norm(167)
thf(fact_4194_semiring__norm_I166_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W: num,Y: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ V2 @ W ) @ Y ) ) ) ).
% semiring_norm(166)
thf(fact_4195_diff__numeral__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ M2 ) ) ) ).
% diff_numeral_simps(4)
thf(fact_4196_rotate1__length01,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) )
=> ( ( rotate1 @ A @ Xs2 )
= Xs2 ) ) ).
% rotate1_length01
thf(fact_4197_sub__num__simps_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K2 ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl_inc @ A @ ( neg_numeral_sub @ A @ K2 @ L ) ) ) ) ).
% sub_num_simps(8)
thf(fact_4198_sub__num__simps_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K2 ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl_dec @ A @ ( neg_numeral_sub @ A @ K2 @ L ) ) ) ) ).
% sub_num_simps(7)
thf(fact_4199_diff__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ M2 @ one2 ) ) ) ).
% diff_numeral_special(2)
thf(fact_4200_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_4201_sub__num__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K2 ) @ one2 )
= ( numeral_numeral @ A @ ( bit0 @ K2 ) ) ) ) ).
% sub_num_simps(5)
thf(fact_4202_not__minus__numeral__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% not_minus_numeral_eq
thf(fact_4203_sub__num__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K2 ) @ one2 )
= ( numeral_numeral @ A @ ( bitM @ K2 ) ) ) ) ).
% sub_num_simps(4)
thf(fact_4204_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_4205_add__neg__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ M2 @ one2 ) ) ) ).
% add_neg_numeral_special(3)
thf(fact_4206_add__neg__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% add_neg_numeral_special(2)
thf(fact_4207_add__neg__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% add_neg_numeral_special(1)
thf(fact_4208_minus__sub__one__diff__one,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( neg_numeral_sub @ A @ M2 @ one2 ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% minus_sub_one_diff_one
thf(fact_4209_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_4210_diff__numeral__special_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% diff_numeral_special(8)
thf(fact_4211_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_4212_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_4213_funpow__swap1,axiom,
! [A: $tType,F3: A > A,N: nat,X3: A] :
( ( F3 @ ( compow @ ( A > A ) @ N @ F3 @ X3 ) )
= ( compow @ ( A > A ) @ N @ F3 @ ( F3 @ X3 ) ) ) ).
% funpow_swap1
thf(fact_4214_funpow__mult,axiom,
! [A: $tType,N: nat,M2: nat,F3: A > A] :
( ( compow @ ( A > A ) @ N @ ( compow @ ( A > A ) @ M2 @ F3 ) )
= ( compow @ ( A > A ) @ ( times_times @ nat @ M2 @ N ) @ F3 ) ) ).
% funpow_mult
thf(fact_4215_bij__betw__funpow,axiom,
! [A: $tType,F3: A > A,S3: set @ A,N: nat] :
( ( bij_betw @ A @ A @ F3 @ S3 @ S3 )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ S3 @ S3 ) ) ).
% bij_betw_funpow
thf(fact_4216_funpow__times__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [F3: A > nat,X3: A] :
( ( compow @ ( A > A ) @ ( F3 @ X3 ) @ ( times_times @ A @ X3 ) )
= ( times_times @ A @ ( power_power @ A @ X3 @ ( F3 @ X3 ) ) ) ) ) ).
% funpow_times_power
thf(fact_4217_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_4218_sub__non__negative,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M2: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M2 ) )
= ( ord_less_eq @ num @ M2 @ N ) ) ) ).
% sub_non_negative
thf(fact_4219_sub__non__positive,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M2: num] :
( ( ord_less_eq @ A @ ( neg_numeral_sub @ A @ N @ M2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ num @ N @ M2 ) ) ) ).
% sub_non_positive
thf(fact_4220_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K2: num,A3: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ K2 ) @ A3 )
= ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K2 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A3 ) ) ) ).
% numeral_add_unfold_funpow
thf(fact_4221_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N3: nat] : ( compow @ ( A > A ) @ N3 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_4222_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_4223_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_4224_minus__numeral__eq__not__sub__one,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( neg_numeral_sub @ A @ N @ one2 ) ) ) ) ).
% minus_numeral_eq_not_sub_one
thf(fact_4225_sub__BitM__One__eq,axiom,
! [N: num] :
( ( neg_numeral_sub @ int @ ( bitM @ N ) @ one2 )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( neg_numeral_sub @ int @ N @ one2 ) ) ) ).
% sub_BitM_One_eq
thf(fact_4226_relpowp__fun__conv,axiom,
! [A: $tType] :
( ( compow @ ( A > A > $o ) )
= ( ^ [N3: nat,P4: A > A > $o,X4: A,Y3: A] :
? [F4: nat > A] :
( ( ( F4 @ ( zero_zero @ nat ) )
= X4 )
& ( ( F4 @ N3 )
= Y3 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N3 )
=> ( P4 @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) ) ) ) ) ).
% relpowp_fun_conv
thf(fact_4227_Nat_Ofunpow__code__def,axiom,
! [A: $tType] :
( ( funpow @ A )
= ( compow @ ( A > A ) ) ) ).
% Nat.funpow_code_def
thf(fact_4228_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X4: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X4 )
@ ^ [X4: A,Y3: A] : ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_4229_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F3: C > B,G3: D > A,X3: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apfst @ D @ A @ C @ G3 @ X3 ) )
= ( product_Pair @ A @ B @ ( G3 @ ( product_fst @ D @ C @ X3 ) ) @ ( F3 @ ( product_snd @ D @ C @ X3 ) ) ) ) ).
% apsnd_apfst
thf(fact_4230_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F3: C > A,X3: C,Y: B] :
( ( product_apfst @ C @ A @ B @ F3 @ ( product_Pair @ C @ B @ X3 @ Y ) )
= ( product_Pair @ A @ B @ ( F3 @ X3 ) @ Y ) ) ).
% apfst_conv
thf(fact_4231_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > A,X3: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F3 @ X3 ) )
= ( F3 @ ( product_fst @ C @ B @ X3 ) ) ) ).
% fst_apfst
thf(fact_4232_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > A,X3: product_prod @ C @ B,G3: C > A] :
( ( ( product_apfst @ C @ A @ B @ F3 @ X3 )
= ( product_apfst @ C @ A @ B @ G3 @ X3 ) )
= ( ( F3 @ ( product_fst @ C @ B @ X3 ) )
= ( G3 @ ( product_fst @ C @ B @ X3 ) ) ) ) ).
% apfst_eq_conv
thf(fact_4233_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F3: C > B,X3: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F3 @ X3 ) )
= ( product_snd @ C @ A @ X3 ) ) ).
% snd_apfst
thf(fact_4234_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F3: C > A,G3: D > B,X3: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F3 @ ( product_apsnd @ D @ B @ C @ G3 @ X3 ) )
= ( product_Pair @ A @ B @ ( F3 @ ( product_fst @ C @ D @ X3 ) ) @ ( G3 @ ( product_snd @ C @ D @ X3 ) ) ) ) ).
% apfst_apsnd
thf(fact_4235_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F3: C > B,G3: D > A,P2: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apfst @ D @ A @ C @ G3 @ P2 ) )
= ( product_apfst @ D @ A @ B @ G3 @ ( product_apsnd @ C @ B @ D @ F3 @ P2 ) ) ) ).
% apsnd_apfst_commute
thf(fact_4236_relpowp__Suc__E,axiom,
! [A: $tType,N: nat,P: A > A > $o,X3: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X3 @ Z2 )
=> ~ ! [Y4: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X3 @ Y4 )
=> ~ ( P @ Y4 @ Z2 ) ) ) ).
% relpowp_Suc_E
thf(fact_4237_relpowp__Suc__I,axiom,
! [A: $tType,N: nat,P: A > A > $o,X3: A,Y: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X3 @ Y )
=> ( ( P @ Y @ Z2 )
=> ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X3 @ Z2 ) ) ) ).
% relpowp_Suc_I
thf(fact_4238_relpowp__Suc__D2,axiom,
! [A: $tType,N: nat,P: A > A > $o,X3: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X3 @ Z2 )
=> ? [Y4: A] :
( ( P @ X3 @ Y4 )
& ( compow @ ( A > A > $o ) @ N @ P @ Y4 @ Z2 ) ) ) ).
% relpowp_Suc_D2
thf(fact_4239_relpowp__Suc__E2,axiom,
! [A: $tType,N: nat,P: A > A > $o,X3: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X3 @ Z2 )
=> ~ ! [Y4: A] :
( ( P @ X3 @ Y4 )
=> ~ ( compow @ ( A > A > $o ) @ N @ P @ Y4 @ Z2 ) ) ) ).
% relpowp_Suc_E2
thf(fact_4240_relpowp__Suc__I2,axiom,
! [A: $tType,P: A > A > $o,X3: A,Y: A,N: nat,Z2: A] :
( ( P @ X3 @ Y )
=> ( ( compow @ ( A > A > $o ) @ N @ P @ Y @ Z2 )
=> ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X3 @ Z2 ) ) ) ).
% relpowp_Suc_I2
thf(fact_4241_relpowp__E,axiom,
! [A: $tType,N: nat,P: A > A > $o,X3: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X3 @ Z2 )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X3 != Z2 ) )
=> ~ ! [Y4: A,M: nat] :
( ( N
= ( suc @ M ) )
=> ( ( compow @ ( A > A > $o ) @ M @ P @ X3 @ Y4 )
=> ~ ( P @ Y4 @ Z2 ) ) ) ) ) ).
% relpowp_E
thf(fact_4242_relpowp__E2,axiom,
! [A: $tType,N: nat,P: A > A > $o,X3: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X3 @ Z2 )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X3 != Z2 ) )
=> ~ ! [Y4: A,M: nat] :
( ( N
= ( suc @ M ) )
=> ( ( P @ X3 @ Y4 )
=> ~ ( compow @ ( A > A > $o ) @ M @ P @ Y4 @ Z2 ) ) ) ) ) ).
% relpowp_E2
thf(fact_4243_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
@ ^ [X4: nat,Y3: nat] : ( ord_less_eq @ nat @ Y3 @ X4 )
@ ^ [X4: nat,Y3: nat] : ( ord_less @ nat @ Y3 @ X4 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_4244_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q3: product_prod @ A @ B,F3: C > A,P2: product_prod @ C @ B] :
( ( Q3
= ( product_apfst @ C @ A @ B @ F3 @ P2 ) )
=> ~ ! [X5: C,Y4: B] :
( ( P2
= ( product_Pair @ C @ B @ X5 @ Y4 ) )
=> ( Q3
!= ( product_Pair @ A @ B @ ( F3 @ X5 ) @ Y4 ) ) ) ) ).
% apfst_convE
thf(fact_4245_set__removeAll,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( set2 @ A @ ( removeAll @ A @ X3 @ Xs2 ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_removeAll
thf(fact_4246_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S3: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ? [X: A] :
( ( member @ A @ X @ S3 )
& ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S3 ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_4247_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S3: set @ A,Y: A,F3: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ Y @ S3 )
=> ( ord_less_eq @ B @ ( F3 @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S3 ) ) @ ( F3 @ Y ) ) ) ) ) ) ).
% arg_min_least
thf(fact_4248_removeAll__id,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( removeAll @ A @ X3 @ Xs2 )
= Xs2 ) ) ).
% removeAll_id
thf(fact_4249_length__removeAll__less__eq,axiom,
! [A: $tType,X3: A,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X3 @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_removeAll_less_eq
thf(fact_4250_length__removeAll__less,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X3 @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_removeAll_less
thf(fact_4251_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S3: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S3 ) @ S3 ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_4252_distinct__concat__iff,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( distinct @ A @ ( concat @ A @ Xs2 ) )
= ( ( distinct @ ( list @ A ) @ ( removeAll @ ( list @ A ) @ ( nil @ A ) @ Xs2 ) )
& ! [Ys3: list @ A] :
( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs2 ) )
=> ( distinct @ A @ Ys3 ) )
& ! [Ys3: list @ A,Zs3: list @ A] :
( ( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs2 ) )
& ( member @ ( list @ A ) @ Zs3 @ ( set2 @ ( list @ A ) @ Xs2 ) )
& ( Ys3 != Zs3 ) )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Zs3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% distinct_concat_iff
thf(fact_4253_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
@ ( comp @ code_integer @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( comp @ ( code_integer > code_integer ) @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( product_apsnd @ code_integer @ code_integer @ code_integer ) @ ( times_times @ code_integer ) ) @ ( sgn_sgn @ code_integer ) @ L2
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( ( sgn_sgn @ code_integer @ K3 )
= ( sgn_sgn @ code_integer @ L2 ) )
@ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S7: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S7
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L2 ) @ S7 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_4254_sndI,axiom,
! [A: $tType,B: $tType,X3: product_prod @ A @ B,Y: A,Z2: B] :
( ( X3
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_snd @ A @ B @ X3 )
= Z2 ) ) ).
% sndI
thf(fact_4255_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P2 ) )
= ( ? [A8: B] :
( P2
= ( product_Pair @ B @ A @ A8 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_4256_set__empty2,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs2 ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_4257_set__empty,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( set2 @ A @ Xs2 )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_4258_length__0__conv,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( zero_zero @ nat ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% length_0_conv
thf(fact_4259_length__greater__0__conv,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= ( Xs2
!= ( nil @ A ) ) ) ).
% length_greater_0_conv
thf(fact_4260_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F3: C > B,G3: D > C,X3: product_prod @ A @ D] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apsnd @ D @ C @ A @ G3 @ X3 ) )
= ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F3 @ G3 ) @ X3 ) ) ).
% apsnd_compose
thf(fact_4261_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G3: D > C,X3: product_prod @ D @ B] :
( ( product_apfst @ C @ A @ B @ F3 @ ( product_apfst @ D @ C @ B @ G3 @ X3 ) )
= ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F3 @ G3 ) @ X3 ) ) ).
% apfst_compose
thf(fact_4262_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_4263_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_4264_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_4265_funpow__add,axiom,
! [A: $tType,M2: nat,N: nat,F3: A > A] :
( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M2 @ N ) @ F3 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M2 @ F3 ) @ ( compow @ ( A > A ) @ N @ F3 ) ) ) ).
% funpow_add
thf(fact_4266_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_4267_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_4268_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( comm_monoid_add @ B )
& ( comm_monoid_add @ A ) )
=> ! [H: B > A,G3: C > B,A6: set @ C] :
( ( ( H @ ( zero_zero @ B ) )
= ( zero_zero @ A ) )
=> ( ! [X5: B,Y4: B] :
( ( H @ ( plus_plus @ B @ X5 @ Y4 ) )
= ( plus_plus @ A @ ( H @ X5 ) @ ( H @ Y4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H @ G3 ) @ A6 )
= ( H @ ( groups7311177749621191930dd_sum @ C @ B @ G3 @ A6 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_4269_sum_Osize__neq,axiom,
! [A: $tType,B: $tType,X3: sum_sum @ A @ B] :
( ( size_size @ ( sum_sum @ A @ B ) @ X3 )
!= ( zero_zero @ nat ) ) ).
% sum.size_neq
thf(fact_4270_prod_Osize__neq,axiom,
! [A: $tType,B: $tType,X3: product_prod @ A @ B] :
( ( size_size @ ( product_prod @ A @ B ) @ X3 )
!= ( zero_zero @ nat ) ) ).
% prod.size_neq
thf(fact_4271_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F3: D > C > A,G3: B > D,X3: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F3 @ G3 ) @ X3 )
= ( F3 @ ( G3 @ ( product_fst @ B @ C @ X3 ) ) @ ( product_snd @ B @ C @ X3 ) ) ) ).
% case_prod_comp
thf(fact_4272_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_4273_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A3: A,P2: product_prod @ A @ B] :
( ( A3
= ( product_fst @ A @ B @ P2 ) )
= ( ? [B8: B] :
( P2
= ( product_Pair @ A @ B @ A3 @ B8 ) ) ) ) ).
% eq_fst_iff
thf(fact_4274_fstI,axiom,
! [B: $tType,A: $tType,X3: product_prod @ A @ B,Y: A,Z2: B] :
( ( X3
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_fst @ A @ B @ X3 )
= Y ) ) ).
% fstI
thf(fact_4275_times__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( times_times @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y3 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V5 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) )
@ Xa2
@ X3 ) ) ) ).
% times_int.abs_eq
thf(fact_4276_insert__subsetI,axiom,
! [A: $tType,X3: A,A6: set @ A,X6: set @ A] :
( ( member @ A @ X3 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X3 @ X6 ) @ A6 ) ) ) ).
% insert_subsetI
thf(fact_4277_eq__numeral__iff__iszero_I7_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X3 @ one2 ) ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_4278_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_4279_fst__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F3 ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F3 @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_apfst
thf(fact_4280_snd__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F3 ) )
= ( product_snd @ A @ B ) ) ).
% snd_comp_apfst
thf(fact_4281_fst__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F3: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 ) )
= ( product_fst @ A @ B ) ) ).
% fst_comp_apsnd
thf(fact_4282_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F3: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ F3 @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_apsnd
thf(fact_4283_iszero__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: num] :
( ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W ) ) ) ) ).
% iszero_neg_numeral
thf(fact_4284_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_4285_fst__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% fst_diag_fst
thf(fact_4286_snd__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X4: B] : ( product_Pair @ B @ B @ X4 @ X4 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% snd_diag_snd
thf(fact_4287_eq__Abs__Integ,axiom,
! [Z2: int] :
~ ! [X5: nat,Y4: nat] :
( Z2
!= ( abs_Integ @ ( product_Pair @ nat @ nat @ X5 @ Y4 ) ) ) ).
% eq_Abs_Integ
thf(fact_4288_fst__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X4: B] : ( product_Pair @ B @ B @ X4 @ X4 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% fst_diag_snd
thf(fact_4289_snd__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% snd_diag_fst
thf(fact_4290_not__iszero__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W: num] :
~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W ) ) ) ).
% not_iszero_numeral
thf(fact_4291_eq__numeral__iff__iszero_I9_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: num] :
( ( ( numeral_numeral @ A @ X3 )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X3 ) ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_4292_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_4293_not__iszero__Numeral1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ one2 ) ) ) ).
% not_iszero_Numeral1
thf(fact_4294_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_Suc_atMost_Suc_shift
thf(fact_4295_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_Suc_lessThan_Suc_shift
thf(fact_4296_sum_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ K2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeastAtMost_shift_bounds
thf(fact_4297_sum_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ K2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeastLessThan_shift_bounds
thf(fact_4298_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_Suc_atMost_Suc_shift
thf(fact_4299_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_Suc_lessThan_Suc_shift
thf(fact_4300_prod_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ K2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeastAtMost_shift_bounds
thf(fact_4301_prod_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ K2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeastLessThan_shift_bounds
thf(fact_4302_bit__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
= ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A3 ) @ ( plus_plus @ nat @ N ) ) ) ) ).
% bit_drop_bit_eq
thf(fact_4303_eq__numeral__iff__iszero_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: num,Y: num] :
( ( ( numeral_numeral @ A @ X3 )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X3 @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(1)
thf(fact_4304_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S: B,R: set @ ( product_prod @ A @ B ),S8: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S ) @ R )
=> ( ( S8 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S8 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_4305_zero__int__def,axiom,
( ( zero_zero @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% zero_int_def
thf(fact_4306_int__def,axiom,
( ( semiring_1_of_nat @ int )
= ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N3 @ ( zero_zero @ nat ) ) ) ) ) ).
% int_def
thf(fact_4307_eq__numeral__iff__iszero_I11_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X3 ) ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_4308_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_4309_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_4310_snd__fst__flip,axiom,
! [A: $tType,B: $tType] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X4: B,Y3: A] : ( product_Pair @ A @ B @ Y3 @ X4 ) ) ) ) ).
% snd_fst_flip
thf(fact_4311_fst__snd__flip,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X4: A,Y3: B] : ( product_Pair @ B @ A @ Y3 @ X4 ) ) ) ) ).
% fst_snd_flip
thf(fact_4312_eq__numeral__iff__iszero_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: num,Y: num] :
( ( ( numeral_numeral @ A @ X3 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X3 @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_4313_eq__numeral__iff__iszero_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: num,Y: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X3 @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_4314_eq__numeral__iff__iszero_I4_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: num,Y: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ Y @ X3 ) ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_4315_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C4: set @ ( set @ B ),G3: B > A] :
( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ C4 )
=> ( finite_finite2 @ B @ X5 ) )
=> ( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ C4 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C4 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X5 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( complete_Sup_Sup @ ( set @ B ) @ C4 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G3 @ C4 ) ) ) ) ) ).
% prod.Union_disjoint
thf(fact_4316_uminus__int_Oabs__eq,axiom,
! [X3: product_prod @ nat @ nat] :
( ( uminus_uminus @ int @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X4 )
@ X3 ) ) ) ).
% uminus_int.abs_eq
thf(fact_4317_Collect__restrict,axiom,
! [A: $tType,X6: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X6 )
& ( P @ X4 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_4318_prop__restrict,axiom,
! [A: $tType,X3: A,Z7: set @ A,X6: set @ A,P: A > $o] :
( ( member @ A @ X3 @ Z7 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z7
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X6 )
& ( P @ X4 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_4319_sum_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc_shift
thf(fact_4320_sum_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_lessThan_Suc_shift
thf(fact_4321_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_4322_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_4323_sum_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% sum.atLeastLessThan_shift_0
thf(fact_4324_prod_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% prod.atLeastLessThan_shift_0
thf(fact_4325_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_4326_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_atMost_pred_shift
thf(fact_4327_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_lessThan_pred_shift
thf(fact_4328_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_atMost_pred_shift
thf(fact_4329_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_lessThan_pred_shift
thf(fact_4330_less__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( ord_less @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) )
@ Xa2
@ X3 ) ) ).
% less_int.abs_eq
thf(fact_4331_less__eq__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( ord_less_eq @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) )
@ Xa2
@ X3 ) ) ).
% less_eq_int.abs_eq
thf(fact_4332_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_4333_eq__numeral__iff__iszero_I5_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: num] :
( ( ( numeral_numeral @ A @ X3 )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X3 @ one2 ) ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_4334_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% sum.atLeastAtMost_shift_0
thf(fact_4335_plus__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( plus_plus @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V5 ) ) )
@ Xa2
@ X3 ) ) ) ).
% plus_int.abs_eq
thf(fact_4336_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% prod.atLeastAtMost_shift_0
thf(fact_4337_minus__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( minus_minus @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) )
@ Xa2
@ X3 ) ) ) ).
% minus_int.abs_eq
thf(fact_4338_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C4: set @ ( set @ B ),G3: B > A] :
( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ C4 )
=> ( finite_finite2 @ B @ X5 ) )
=> ( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ C4 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C4 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X5 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( complete_Sup_Sup @ ( set @ B ) @ C4 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G3 @ C4 ) ) ) ) ) ).
% sum.Union_disjoint
thf(fact_4339_subset__emptyI,axiom,
! [A: $tType,A6: set @ A] :
( ! [X5: A] :
~ ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_4340_Code__Numeral_Onegative__def,axiom,
( code_negative
= ( comp @ code_integer @ code_integer @ num @ ( uminus_uminus @ code_integer ) @ ( numeral_numeral @ code_integer ) ) ) ).
% Code_Numeral.negative_def
thf(fact_4341_Code__Target__Int_Onegative__def,axiom,
( code_Target_negative
= ( comp @ int @ int @ num @ ( uminus_uminus @ int ) @ ( numeral_numeral @ int ) ) ) ).
% Code_Target_Int.negative_def
thf(fact_4342_pred__nat__def,axiom,
( pred_nat
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [M5: nat,N3: nat] :
( N3
= ( suc @ M5 ) ) ) ) ) ).
% pred_nat_def
thf(fact_4343_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_4344_num__of__nat__numeral__eq,axiom,
! [Q3: num] :
( ( num_of_nat @ ( numeral_numeral @ nat @ Q3 ) )
= Q3 ) ).
% num_of_nat_numeral_eq
thf(fact_4345_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ ( zero_zero @ nat ) )
= one2 ) ).
% num_of_nat.simps(1)
thf(fact_4346_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_4347_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_4348_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_4349_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_4350_num__of__nat__plus__distrib,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( plus_plus @ nat @ M2 @ N ) )
= ( plus_plus @ num @ ( num_of_nat @ M2 ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_4351_less__eq__int_Orep__eq,axiom,
( ( ord_less_eq @ int )
= ( ^ [X4: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y3: nat,Z4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y3 @ V5 ) @ ( plus_plus @ nat @ U2 @ Z4 ) ) )
@ ( rep_Integ @ X4 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_4352_less__int_Orep__eq,axiom,
( ( ord_less @ int )
= ( ^ [X4: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y3: nat,Z4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y3 @ V5 ) @ ( plus_plus @ nat @ U2 @ Z4 ) ) )
@ ( rep_Integ @ X4 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_4353_lex__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( lex_prod @ A @ B )
= ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [A8: A,B8: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A14: A,B12: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ A14 ) @ Ra )
| ( ( A8 = A14 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B8 @ B12 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_4354_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,P2: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( P2 @ X4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P2 @ ( insert2 @ B @ I @ I5 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P2 @ I5 ) ) )
& ( ~ ( member @ B @ I @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P2 @ ( insert2 @ B @ I @ I5 ) )
= ( times_times @ A @ ( P2 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P2 @ I5 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_4355_in__lex__prod,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Pair @ A @ B @ A4 @ B3 ) ) @ ( lex_prod @ A @ B @ R2 @ S ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A4 ) @ R2 )
| ( ( A3 = A4 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_4356_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P2: B > A] :
( ( groups1962203154675924110t_prod @ B @ A @ P2 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty'
thf(fact_4357_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T4: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G3 @ S3 )
= ( groups1962203154675924110t_prod @ B @ A @ G3 @ T4 ) ) ) ) ) ).
% prod.mono_neutral_left'
thf(fact_4358_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T4: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G3 @ T4 )
= ( groups1962203154675924110t_prod @ B @ A @ G3 @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right'
thf(fact_4359_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T4: set @ B,H: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( H @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G3 @ X5 )
= ( H @ X5 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G3 @ S3 )
= ( groups1962203154675924110t_prod @ B @ A @ H @ T4 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
thf(fact_4360_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T4: set @ B,G3: B > A,H: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X5 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G3 @ X5 )
= ( H @ X5 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G3 @ T4 )
= ( groups1962203154675924110t_prod @ B @ A @ H @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
thf(fact_4361_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P4: A > $o,R6: A > ( set @ ( product_prod @ B @ B ) )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [X9: A,Y7: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y3: B] :
( ( X9 = X4 )
& ( P4 @ X4 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y3 ) @ ( R6 @ X4 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_4362_uminus__int__def,axiom,
( ( uminus_uminus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X4 ) ) ) ) ).
% uminus_int_def
thf(fact_4363_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( remove1 @ A @ X3 @ ( linord4507533701916653071of_set @ A @ A6 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_4364_in__set__product__lists__length,axiom,
! [A: $tType,Xs2: list @ A,Xss: list @ ( list @ A )] :
( ( member @ ( list @ A ) @ Xs2 @ ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss ) ) )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ ( list @ A ) ) @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_4365_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_4366_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( set2 @ A @ ( linord4507533701916653071of_set @ A @ A6 ) )
= A6 ) ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_4367_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( size_size @ ( list @ A ) @ ( linord4507533701916653071of_set @ A @ A6 ) )
= ( finite_card @ A @ A6 ) ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_4368_same__fstI,axiom,
! [B: $tType,A: $tType,P: A > $o,X3: A,Y9: B,Y: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P @ X3 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y9 @ Y ) @ ( R @ X3 ) )
=> ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y9 ) @ ( product_Pair @ A @ B @ X3 @ Y ) ) @ ( same_fst @ A @ B @ P @ R ) ) ) ) ).
% same_fstI
thf(fact_4369_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( linord4507533701916653071of_set @ A @ A6 )
= ( nil @ A ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_4370_times__int__def,axiom,
( ( times_times @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y3 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V5 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_4371_minus__int__def,axiom,
( ( minus_minus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_4372_plus__int__def,axiom,
( ( plus_plus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V5 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_4373_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_4374_length__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Xs2: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs2 ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ).
% length_insort
thf(fact_4375_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X3 @ A6 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ ( linord4507533701916653071of_set @ A @ A6 ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_4376_set__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Xs2: list @ B] :
( ( set2 @ B @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs2 ) )
= ( insert2 @ B @ X3 @ ( set2 @ B @ Xs2 ) ) ) ) ).
% set_insort_key
thf(fact_4377_distinct__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Xs2: list @ B] :
( ( distinct @ B @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs2 ) )
= ( ~ ( member @ B @ X3 @ ( set2 @ B @ Xs2 ) )
& ( distinct @ B @ Xs2 ) ) ) ) ).
% distinct_insort
thf(fact_4378_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( linord4507533701916653071of_set @ A @ A6 )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_4379_Gcd__remove0__nat,axiom,
! [M7: set @ nat] :
( ( finite_finite2 @ nat @ M7 )
=> ( ( gcd_Gcd @ nat @ M7 )
= ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M7 @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_4380_integer__of__num__triv_I2_J,axiom,
( ( code_integer_of_num @ ( bit0 @ one2 ) )
= ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ).
% integer_of_num_triv(2)
thf(fact_4381_take__bit__numeral__minus__numeral__int,axiom,
! [M2: num,N: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int )
@ ^ [Q4: num] : ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M2 ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ M2 ) ) @ ( numeral_numeral @ int @ Q4 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M2 ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_4382_image__minus__const__atLeastLessThan__nat,axiom,
! [C3: nat,Y: nat,X3: nat] :
( ( ( ord_less @ nat @ C3 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X3 @ C3 ) @ ( minus_minus @ nat @ Y @ C3 ) ) ) )
& ( ~ ( ord_less @ nat @ C3 @ Y )
=> ( ( ( ord_less @ nat @ X3 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X3 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_4383_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A,X3: B,A6: set @ B] :
( ( B2
= ( F3 @ X3 ) )
=> ( ( member @ B @ X3 @ A6 )
=> ( member @ A @ B2 @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ).
% image_eqI
thf(fact_4384_image__ident,axiom,
! [A: $tType,Y8: set @ A] :
( ( image2 @ A @ A
@ ^ [X4: A] : X4
@ Y8 )
= Y8 ) ).
% image_ident
thf(fact_4385_image__is__empty,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B] :
( ( ( image2 @ B @ A @ F3 @ A6 )
= ( bot_bot @ ( set @ A ) ) )
= ( A6
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_4386_empty__is__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image2 @ B @ A @ F3 @ A6 ) )
= ( A6
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_4387_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_4388_insert__image,axiom,
! [B: $tType,A: $tType,X3: A,A6: set @ A,F3: A > B] :
( ( member @ A @ X3 @ A6 )
=> ( ( insert2 @ B @ ( F3 @ X3 ) @ ( image2 @ A @ B @ F3 @ A6 ) )
= ( image2 @ A @ B @ F3 @ A6 ) ) ) ).
% insert_image
thf(fact_4389_image__insert,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: B,B5: set @ B] :
( ( image2 @ B @ A @ F3 @ ( insert2 @ B @ A3 @ B5 ) )
= ( insert2 @ A @ ( F3 @ A3 ) @ ( image2 @ B @ A @ F3 @ B5 ) ) ) ).
% image_insert
thf(fact_4390_take__bit__num__simps_I1_J,axiom,
! [M2: num] :
( ( bit_take_bit_num @ ( zero_zero @ nat ) @ M2 )
= ( none @ num ) ) ).
% take_bit_num_simps(1)
thf(fact_4391_bij__betw__Suc,axiom,
! [M7: set @ nat,N5: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M7 @ N5 )
= ( ( image2 @ nat @ nat @ suc @ M7 )
= N5 ) ) ).
% bij_betw_Suc
thf(fact_4392_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S3 )
= S3 ) ) ).
% image_add_0
thf(fact_4393_image__add__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I: A,J: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K2 ) @ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ K2 ) ) ) ) ).
% image_add_atLeastAtMost
thf(fact_4394_image__add__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I: A,J: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K2 ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ K2 ) ) ) ) ).
% image_add_atLeastLessThan
thf(fact_4395_image__add__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [C3: A,A3: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ C3 ) @ ( set_ord_atMost @ A @ A3 ) )
= ( set_ord_atMost @ A @ ( plus_plus @ A @ C3 @ A3 ) ) ) ) ).
% image_add_atMost
thf(fact_4396_bij__betw__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A6: set @ A,B5: set @ A] :
( ( bij_betw @ A @ A @ ( plus_plus @ A @ A3 ) @ A6 @ B5 )
= ( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ A6 )
= B5 ) ) ) ).
% bij_betw_add
thf(fact_4397_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_4398_take__bit__num__simps_I2_J,axiom,
! [N: nat] :
( ( bit_take_bit_num @ ( suc @ N ) @ one2 )
= ( some @ num @ one2 ) ) ).
% take_bit_num_simps(2)
thf(fact_4399_take__bit__num__simps_I5_J,axiom,
! [R2: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R2 ) @ one2 )
= ( some @ num @ one2 ) ) ).
% take_bit_num_simps(5)
thf(fact_4400_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
= ( set_or1337092689740270186AtMost @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_4401_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) )
= ( set_or7035219750837199246ssThan @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_4402_ccSUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( bot_bot @ A )
@ A6 ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_bot
thf(fact_4403_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( bot_bot @ A )
@ A6 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_4404_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: B > A,A6: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B5 @ A6 ) )
= ( bot_bot @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ( B5 @ X4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_4405_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: B > A,A6: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B5 @ A6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ( B5 @ X4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_4406_ccSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,F3: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : F3
@ A6 ) )
= F3 ) ) ) ).
% ccSUP_const
thf(fact_4407_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,C3: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : C3
@ A6 ) )
= C3 ) ) ) ).
% cSUP_const
thf(fact_4408_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F3: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : F3
@ A6 ) )
= F3 ) ) ) ).
% SUP_const
thf(fact_4409_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I: A,J: A] :
( ( image2 @ A @ A
@ ^ [N3: A] : ( plus_plus @ A @ N3 @ K2 )
@ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ K2 ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_4410_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,C3: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : C3
@ A6 ) )
= C3 ) ) ) ).
% cINF_const
thf(fact_4411_INF__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F3: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : F3
@ A6 ) )
= F3 ) ) ) ).
% INF_const
thf(fact_4412_ccINF__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,F3: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : F3
@ A6 ) )
= F3 ) ) ) ).
% ccINF_const
thf(fact_4413_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I: A,J: A] :
( ( image2 @ A @ A
@ ^ [N3: A] : ( plus_plus @ A @ N3 @ K2 )
@ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ K2 ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_4414_if__image__distrib,axiom,
! [A: $tType,B: $tType,P: B > $o,F3: B > A,G3: B > A,S3: set @ B] :
( ( image2 @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ S3 )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ S3 @ ( collect @ B @ P ) ) )
@ ( image2 @ B @ A @ G3
@ ( inf_inf @ ( set @ B ) @ S3
@ ( collect @ B
@ ^ [X4: B] :
~ ( P @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_4415_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A6: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X4 )
=> ? [Y3: B] :
( ( member @ B @ Y3 @ A6 )
& ( ord_less @ A @ ( F3 @ Y3 ) @ X4 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_4416_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_4417_Gcd__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( ( gcd_Gcd @ A @ A6 )
= ( zero_zero @ A ) )
= ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_4418_take__bit__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: num,N: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M2 ) @ N ) ) ) ) ).
% take_bit_numeral_numeral
thf(fact_4419_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert2 @ A @ ( F3 @ X4 ) @ ( bot_bot @ ( set @ A ) ) )
@ A6 ) )
= ( image2 @ B @ A @ F3 @ A6 ) ) ).
% UNION_singleton_eq_range
thf(fact_4420_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: A > B,B5: set @ B] :
( ! [X5: A] :
( ( P @ X5 )
=> ( member @ B @ ( F3 @ X5 ) @ B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( collect @ A @ P ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_4421_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F3: B > A,A6: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A6 ) @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F3 ) @ ( pow2 @ B @ A6 ) ) @ ( pow2 @ A @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_4422_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F3: B > A,A6: set @ B,B5: set @ A] :
( ( ( image2 @ B @ A @ F3 @ A6 )
= B5 )
=> ( ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F3 ) @ ( pow2 @ B @ A6 ) )
= ( pow2 @ A @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_4423_imageI,axiom,
! [B: $tType,A: $tType,X3: A,A6: set @ A,F3: A > B] :
( ( member @ A @ X3 @ A6 )
=> ( member @ B @ ( F3 @ X3 ) @ ( image2 @ A @ B @ F3 @ A6 ) ) ) ).
% imageI
thf(fact_4424_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F3: B > A,A6: set @ B] :
( ( member @ A @ Z2 @ ( image2 @ B @ A @ F3 @ A6 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( Z2
= ( F3 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_4425_bex__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B,P: A > $o] :
( ? [X: A] :
( ( member @ A @ X @ ( image2 @ B @ A @ F3 @ A6 ) )
& ( P @ X ) )
=> ? [X5: B] :
( ( member @ B @ X5 @ A6 )
& ( P @ ( F3 @ X5 ) ) ) ) ).
% bex_imageD
thf(fact_4426_image__cong,axiom,
! [B: $tType,A: $tType,M7: set @ A,N5: set @ A,F3: A > B,G3: A > B] :
( ( M7 = N5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ N5 )
=> ( ( F3 @ X5 )
= ( G3 @ X5 ) ) )
=> ( ( image2 @ A @ B @ F3 @ M7 )
= ( image2 @ A @ B @ G3 @ N5 ) ) ) ) ).
% image_cong
thf(fact_4427_ball__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B,P: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( P @ X5 ) )
=> ! [X: B] :
( ( member @ B @ X @ A6 )
=> ( P @ ( F3 @ X ) ) ) ) ).
% ball_imageD
thf(fact_4428_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X3: A,A6: set @ A,B2: B,F3: A > B] :
( ( member @ A @ X3 @ A6 )
=> ( ( B2
= ( F3 @ X3 ) )
=> ( member @ B @ B2 @ ( image2 @ A @ B @ F3 @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_4429_image__Un,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B,B5: set @ B] :
( ( image2 @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A6 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A6 ) @ ( image2 @ B @ A @ F3 @ B5 ) ) ) ).
% image_Un
thf(fact_4430_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A,A6: set @ B] :
( ( member @ A @ B2 @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ~ ! [X5: B] :
( ( B2
= ( F3 @ X5 ) )
=> ~ ( member @ B @ X5 @ A6 ) ) ) ).
% imageE
thf(fact_4431_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,G3: C > B,A6: set @ C] :
( ( image2 @ B @ A @ F3 @ ( image2 @ C @ B @ G3 @ A6 ) )
= ( image2 @ C @ A
@ ^ [X4: C] : ( F3 @ ( G3 @ X4 ) )
@ A6 ) ) ).
% image_image
thf(fact_4432_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( image2 @ B @ A @ F3 @ A6 ) )
& ( P @ X4 ) ) )
= ( image2 @ B @ A @ F3
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( P @ ( F3 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_4433_option_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H: B > C,F1: B,F22: A > B,Option: option @ A] :
( ( H @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( case_option @ C @ A @ ( H @ F1 )
@ ^ [X4: A] : ( H @ ( F22 @ X4 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_4434_image__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ A,F3: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A6 ) @ ( image2 @ A @ B @ F3 @ B5 ) ) ) ).
% image_mono
thf(fact_4435_image__subsetI,axiom,
! [A: $tType,B: $tType,A6: set @ A,F3: A > B,B5: set @ B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( member @ B @ ( F3 @ X5 ) @ B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A6 ) @ B5 ) ) ).
% image_subsetI
thf(fact_4436_subset__imageE,axiom,
! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ~ ! [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A6 )
=> ( B5
!= ( image2 @ B @ A @ F3 @ C7 ) ) ) ) ).
% subset_imageE
thf(fact_4437_image__subset__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A6 ) @ B5 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( member @ A @ ( F3 @ X4 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_4438_subset__image__iff,axiom,
! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image2 @ B @ A @ F3 @ A6 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A6 )
& ( B5
= ( image2 @ B @ A @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_4439_all__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B,P: ( set @ A ) > $o] :
( ( ! [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A6 )
=> ( P @ ( image2 @ B @ A @ F3 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_4440_option_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > B,X2: A] :
( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X2 ) )
= ( F22 @ X2 ) ) ).
% option.simps(5)
thf(fact_4441_option_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: A > B] :
( ( case_option @ B @ A @ F1 @ F22 @ ( none @ A ) )
= F1 ) ).
% option.simps(4)
thf(fact_4442_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B5: set @ C,F3: B > A,G3: C > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ? [X: C] :
( ( member @ C @ X @ B5 )
& ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G3 @ X ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B5 )
=> ? [X: B] :
( ( member @ B @ X @ A6 )
& ( ord_less_eq @ A @ ( G3 @ J2 ) @ ( F3 @ X ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G3 @ B5 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_4443_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B5: set @ C,G3: C > A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ? [X: C] :
( ( member @ C @ X @ B5 )
& ( ord_less_eq @ A @ ( G3 @ X ) @ ( F3 @ I3 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B5 )
=> ? [X: B] :
( ( member @ B @ X @ A6 )
& ( ord_less_eq @ A @ ( F3 @ X ) @ ( G3 @ J2 ) ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G3 @ B5 ) ) ) ) ) ) ).
% INF_eq
thf(fact_4444_zero__notin__Suc__image,axiom,
! [A6: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ A6 ) ) ).
% zero_notin_Suc_image
thf(fact_4445_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
! [N: nat] :
( ( bit_take_bit_num @ N @ one2 )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N3: nat] : ( some @ num @ one2 )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_4446_all__finite__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B,P: ( set @ A ) > $o] :
( ( ! [B6: set @ A] :
( ( ( finite_finite2 @ A @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F3 @ A6 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set @ B] :
( ( ( finite_finite2 @ B @ B6 )
& ( ord_less_eq @ ( set @ B ) @ B6 @ A6 ) )
=> ( P @ ( image2 @ B @ A @ F3 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_4447_ex__finite__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B,P: ( set @ A ) > $o] :
( ( ? [B6: set @ A] :
( ( finite_finite2 @ A @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F3 @ A6 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set @ B] :
( ( finite_finite2 @ B @ B6 )
& ( ord_less_eq @ ( set @ B ) @ B6 @ A6 )
& ( P @ ( image2 @ B @ A @ F3 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_4448_finite__subset__image,axiom,
! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ? [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A6 )
& ( finite_finite2 @ B @ C7 )
& ( B5
= ( image2 @ B @ A @ F3 @ C7 ) ) ) ) ) ).
% finite_subset_image
thf(fact_4449_finite__surj,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( image2 @ A @ B @ F3 @ A6 ) )
=> ( finite_finite2 @ B @ B5 ) ) ) ).
% finite_surj
thf(fact_4450_translation__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S: set @ A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= ( inf_inf @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_Int
thf(fact_4451_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F3: B > A,X3: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ( F3 @ I3 )
= X3 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ I5 ) )
= X3 ) ) ) ) ).
% SUP_eq_const
thf(fact_4452_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S: set @ A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
= ( minus_minus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_diff
thf(fact_4453_INF__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F3: B > A,X3: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ( F3 @ I3 )
= X3 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I5 ) )
= X3 ) ) ) ) ).
% INF_eq_const
thf(fact_4454_image__Int__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B,B5: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A6 @ B5 ) ) @ ( inf_inf @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A6 ) @ ( image2 @ B @ A @ F3 @ B5 ) ) ) ).
% image_Int_subset
thf(fact_4455_image__diff__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B,B5: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A6 ) @ ( image2 @ B @ A @ F3 @ B5 ) ) @ ( image2 @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A6 @ B5 ) ) ) ).
% image_diff_subset
thf(fact_4456_bij__betw__byWitness,axiom,
! [A: $tType,B: $tType,A6: set @ A,F8: B > A,F3: A > B,A11: set @ B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ( F8 @ ( F3 @ X5 ) )
= X5 ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A11 )
=> ( ( F3 @ ( F8 @ X5 ) )
= X5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A6 ) @ A11 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F8 @ A11 ) @ A6 )
=> ( bij_betw @ A @ B @ F3 @ A6 @ A11 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_4457_bij__betw__subset,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ A,A11: set @ B,B5: set @ A,B13: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A6 @ A11 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ( ( image2 @ A @ B @ F3 @ B5 )
= B13 )
=> ( bij_betw @ A @ B @ F3 @ B5 @ B13 ) ) ) ) ).
% bij_betw_subset
thf(fact_4458_translation__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_Compl
thf(fact_4459_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F3: B > A,X3: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ X3 ) )
=> ( ! [Y4: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ Y4 ) )
=> ( ord_less_eq @ A @ X3 @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
= X3 ) ) ) ) ).
% SUP_eqI
thf(fact_4460_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B5: set @ C,F3: B > A,G3: C > A] :
( ! [N2: B] :
( ( member @ B @ N2 @ A6 )
=> ? [X: C] :
( ( member @ C @ X @ B5 )
& ( ord_less_eq @ A @ ( F3 @ N2 ) @ ( G3 @ X ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G3 @ B5 ) ) ) ) ) ).
% SUP_mono
thf(fact_4461_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F3: B > A,U: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ U ) ) ) ).
% SUP_least
thf(fact_4462_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,G3: B > A,A6: set @ B] :
( ! [X5: B] : ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ A6 ) ) ) ) ) ).
% SUP_mono'
thf(fact_4463_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A6: set @ B,F3: B > A] :
( ( member @ B @ I @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ I ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ).
% SUP_upper
thf(fact_4464_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A6: set @ B,U: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ U )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ U ) ) ) ) ) ).
% SUP_le_iff
thf(fact_4465_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A6: set @ B,U: A,F3: B > A] :
( ( member @ B @ I @ A6 )
=> ( ( ord_less_eq @ A @ U @ ( F3 @ I ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_4466_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,X3: A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ord_less_eq @ A @ X3 @ ( F3 @ I3 ) ) )
=> ( ! [Y4: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A6 )
=> ( ord_less_eq @ A @ Y4 @ ( F3 @ I2 ) ) )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
= X3 ) ) ) ) ).
% INF_eqI
thf(fact_4467_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: set @ B,A6: set @ C,F3: C > A,G3: B > A] :
( ! [M: B] :
( ( member @ B @ M @ B5 )
=> ? [X: C] :
( ( member @ C @ X @ A6 )
& ( ord_less_eq @ A @ ( F3 @ X ) @ ( G3 @ M ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F3 @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B5 ) ) ) ) ) ).
% INF_mono
thf(fact_4468_INF__lower,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A6: set @ B,F3: B > A] :
( ( member @ B @ I @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( F3 @ I ) ) ) ) ).
% INF_lower
thf(fact_4469_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,G3: B > A,A6: set @ B] :
( ! [X5: B] : ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ A6 ) ) ) ) ) ).
% INF_mono'
thf(fact_4470_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A6: set @ B,F3: B > A,U: A] :
( ( member @ B @ I @ A6 )
=> ( ( ord_less_eq @ A @ ( F3 @ I ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ U ) ) ) ) ).
% INF_lower2
thf(fact_4471_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,F3: B > A,A6: set @ B] :
( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ord_less_eq @ A @ U @ ( F3 @ X4 ) ) ) ) ) ) ).
% le_INF_iff
thf(fact_4472_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,U: A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ord_less_eq @ A @ U @ ( F3 @ I3 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ).
% INF_greatest
thf(fact_4473_image__constant__conv,axiom,
! [B: $tType,A: $tType,A6: set @ B,C3: A] :
( ( ( A6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X4: B] : C3
@ A6 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X4: B] : C3
@ A6 )
= ( insert2 @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_4474_image__constant,axiom,
! [A: $tType,B: $tType,X3: A,A6: set @ A,C3: B] :
( ( member @ A @ X3 @ A6 )
=> ( ( image2 @ A @ B
@ ^ [X4: A] : C3
@ A6 )
= ( insert2 @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_4475_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A6: set @ A,F3: A > B,X3: A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ A6 )
=> ( ( F3 @ Y4 )
= ( F3 @ X3 ) ) )
=> ( ( the_elem @ B @ ( image2 @ A @ B @ F3 @ A6 ) )
= ( F3 @ X3 ) ) ) ) ).
% the_elem_image_unique
thf(fact_4476_take__bit__num__eq__Some__imp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: num,Q3: num] :
( ( ( bit_take_bit_num @ M2 @ N )
= ( some @ num @ Q3 ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ Q3 ) ) ) ) ).
% take_bit_num_eq_Some_imp
thf(fact_4477_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X3: A,F3: B > A,A6: set @ B] :
( ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) )
= ( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X3 )
=> ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( ord_less @ A @ Y3 @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_4478_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A6: set @ B,X3: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ X3 )
= ( ! [Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( ord_less @ A @ ( F3 @ X4 ) @ Y3 ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_4479_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F3: B > A,M7: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ M7 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ M7 ) ) ) ) ).
% cSUP_least
thf(fact_4480_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,C3: A,F3: B > A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ord_less_eq @ A @ C3 @ ( F3 @ I3 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ I5 ) )
= C3 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( ( F3 @ X4 )
= C3 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_4481_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,M2: A,F3: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( ord_less_eq @ A @ M2 @ ( F3 @ X5 ) ) )
=> ( ord_less_eq @ A @ M2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_4482_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F3: B > A,C3: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ C3 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I5 ) )
= C3 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( ( F3 @ X4 )
= C3 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4483_integer__of__num__def,axiom,
( code_integer_of_num
= ( numeral_numeral @ code_integer ) ) ).
% integer_of_num_def
thf(fact_4484_card__image__le,axiom,
! [B: $tType,A: $tType,A6: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A6 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( image2 @ A @ B @ F3 @ A6 ) ) @ ( finite_card @ A @ A6 ) ) ) ).
% card_image_le
thf(fact_4485_bij__betw__comp__iff2,axiom,
! [C: $tType,A: $tType,B: $tType,F8: A > B,A11: set @ A,A15: set @ B,F3: C > A,A6: set @ C] :
( ( bij_betw @ A @ B @ F8 @ A11 @ A15 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ F3 @ A6 ) @ A11 )
=> ( ( bij_betw @ C @ A @ F3 @ A6 @ A11 )
= ( bij_betw @ C @ B @ ( comp @ A @ B @ C @ F8 @ F3 ) @ A6 @ A15 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_4486_option_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_option @ B @ A )
= ( ^ [F12: B,F23: A > B,Option3: option @ A] :
( if @ B
@ ( Option3
= ( none @ A ) )
@ F12
@ ( F23 @ ( the2 @ A @ Option3 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_4487_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B5: set @ B,F3: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A6 @ B5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ B5 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_4488_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: set @ B,A6: set @ B,F3: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ A6 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ B5 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B5 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_4489_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,C3: A] :
( ( ( A6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C3
@ A6 ) )
= ( bot_bot @ A ) ) )
& ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C3
@ A6 ) )
= C3 ) ) ) ) ).
% SUP_constant
thf(fact_4490_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_4491_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T4: set @ C,G3: B > C,H: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G3 @ S3 ) @ T4 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y3: C] :
( groups7311177749621191930dd_sum @ B @ A @ H
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S3 )
& ( ( G3 @ X4 )
= Y3 ) ) ) )
@ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ S3 ) ) ) ) ) ) ).
% sum.group
thf(fact_4492_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,X3: A,F3: B > A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : ( inf_inf @ A @ X3 @ ( F3 @ I4 ) )
@ I5 ) )
= ( inf_inf @ A @ X3 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I5 ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_4493_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F3: B > A,X3: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : ( inf_inf @ A @ ( F3 @ I4 ) @ X3 )
@ I5 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I5 ) ) @ X3 ) ) ) ) ).
% INF_inf_const2
thf(fact_4494_SUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A3: B,A6: set @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( insert2 @ B @ A3 @ A6 ) ) )
= ( sup_sup @ A @ ( F3 @ A3 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ).
% SUP_insert
thf(fact_4495_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T4: set @ C,G3: B > C,H: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G3 @ S3 ) @ T4 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y3: C] :
( groups7121269368397514597t_prod @ B @ A @ H
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S3 )
& ( ( G3 @ X4 )
= Y3 ) ) ) )
@ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ S3 ) ) ) ) ) ) ).
% prod.group
thf(fact_4496_INF__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A3: B,A6: set @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( insert2 @ B @ A3 @ A6 ) ) )
= ( inf_inf @ A @ ( F3 @ A3 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ).
% INF_insert
thf(fact_4497_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F3: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_4498_surj__card__le,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( image2 @ A @ B @ F3 @ A6 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B5 ) @ ( finite_card @ A @ A6 ) ) ) ) ).
% surj_card_le
thf(fact_4499_take__bit__num__eq__None__imp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: num] :
( ( ( bit_take_bit_num @ M2 @ N )
= ( none @ num ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( numeral_numeral @ A @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_num_eq_None_imp
thf(fact_4500_image__Suc__lessThan,axiom,
! [N: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ).
% image_Suc_lessThan
thf(fact_4501_image__Suc__atMost,axiom,
! [N: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N ) ) ) ).
% image_Suc_atMost
thf(fact_4502_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_4503_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_4504_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_4505_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_4506_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P @ F1 ) )
| ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
& ~ ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_4507_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ( ( Option
= ( none @ A ) )
=> ( P @ F1 ) )
& ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
=> ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_4508_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I5: set @ C,G3: A > B,F3: C > A] :
( ( finite_finite2 @ C @ I5 )
=> ( ! [I3: C] :
( ( member @ C @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G3 @ ( F3 @ I3 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G3 @ ( image2 @ C @ A @ F3 @ I5 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G3 @ F3 ) @ I5 ) ) ) ) ) ).
% sum_image_le
thf(fact_4509_integer__of__num__triv_I1_J,axiom,
( ( code_integer_of_num @ one2 )
= ( one_one @ code_integer ) ) ).
% integer_of_num_triv(1)
thf(fact_4510_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,X3: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ X3 ) @ ( times_times @ A @ C3 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ Y ) @ ( times_times @ A @ C3 @ X3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ X3 @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_4511_integer__of__num_I2_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit0 @ N ) )
= ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% integer_of_num(2)
thf(fact_4512_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N3: nat,M5: num] :
( if @ ( option @ num )
@ ( ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( numeral_numeral @ nat @ M5 ) )
= ( zero_zero @ nat ) )
@ ( none @ num )
@ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( numeral_numeral @ nat @ M5 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_4513_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A,C3: A] :
( ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X3 @ C3 ) @ ( times_times @ A @ Y @ C3 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C3 ) @ ( times_times @ A @ X3 @ C3 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X3 @ Y )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_4514_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M2: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B2 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A3 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_4515_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M2: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A3 ) @ C3 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B2 ) @ C3 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A3 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_4516_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M2: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C3 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C3 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_4517_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M2: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_4518_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S3: set @ A,R: set @ B,G3: A > B,F3: B > C] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ R )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G3 @ S3 ) @ R )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X4: A] : ( F3 @ ( G3 @ X4 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y3: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S3 )
& ( ( G3 @ X4 )
= Y3 ) ) ) ) )
@ ( F3 @ Y3 ) )
@ R ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_4519_and__minus__numerals_I3_J,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(3)
thf(fact_4520_and__minus__numerals_I7_J,axiom,
! [N: num,M2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M2 ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(7)
thf(fact_4521_and__minus__numerals_I4_J,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(4)
thf(fact_4522_and__minus__numerals_I8_J,axiom,
! [N: num,M2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M2 ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(8)
thf(fact_4523_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,A6: set @ ( product_prod @ A @ B ),F3: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ A6 )
=> ( member @ C @ ( F3 @ A3 @ B2 ) @ ( image2 @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F3 ) @ A6 ) ) ) ).
% pair_imageI
thf(fact_4524_UN__constant,axiom,
! [B: $tType,A: $tType,A6: set @ B,C3: set @ A] :
( ( ( A6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C3
@ A6 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C3
@ A6 ) )
= C3 ) ) ) ).
% UN_constant
thf(fact_4525_UN__singleton,axiom,
! [A: $tType,A6: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ A @ ( set @ A )
@ ^ [X4: A] : ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
@ A6 ) )
= A6 ) ).
% UN_singleton
thf(fact_4526_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C4: set @ B,A3: A,B5: B > ( set @ A )] :
( ( ( C4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert2 @ A @ A3 @ ( B5 @ X4 ) )
@ C4 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert2 @ A @ A3 @ ( B5 @ X4 ) )
@ C4 ) )
= ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ C4 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_4527_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C4: set @ C,A6: C > ( set @ D ),B5: set @ D] :
( ( ( C4
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A6 @ X4 ) @ B5 )
@ C4 ) )
= ( bot_bot @ ( set @ D ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A6 @ X4 ) @ B5 )
@ C4 ) )
= ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A6 @ C4 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_4528_UN__simps_I3_J,axiom,
! [E: $tType,F: $tType,C4: set @ F,A6: set @ E,B5: F > ( set @ E )] :
( ( ( C4
= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F @ ( set @ E )
@ ^ [X4: F] : ( sup_sup @ ( set @ E ) @ A6 @ ( B5 @ X4 ) )
@ C4 ) )
= ( bot_bot @ ( set @ E ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F @ ( set @ E )
@ ^ [X4: F] : ( sup_sup @ ( set @ E ) @ A6 @ ( B5 @ X4 ) )
@ C4 ) )
= ( sup_sup @ ( set @ E ) @ A6 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F @ ( set @ E ) @ B5 @ C4 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_4529_UN__insert,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A3: B,A6: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ ( insert2 @ B @ A3 @ A6 ) ) )
= ( sup_sup @ ( set @ A ) @ ( B5 @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% UN_insert
thf(fact_4530_INT__insert,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A3: B,A6: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ ( insert2 @ B @ A3 @ A6 ) ) )
= ( inf_inf @ ( set @ A ) @ ( B5 @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% INT_insert
thf(fact_4531_set__concat,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( set2 @ A @ ( concat @ A @ Xs2 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xs2 ) ) ) ) ).
% set_concat
thf(fact_4532_take__bit__num__simps_I4_J,axiom,
! [N: nat,M2: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M2 ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N @ M2 ) ) ) ) ).
% take_bit_num_simps(4)
thf(fact_4533_take__bit__num__simps_I3_J,axiom,
! [N: nat,M2: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M2 ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q4: num] : ( some @ num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ N @ M2 ) ) ) ).
% take_bit_num_simps(3)
thf(fact_4534_take__bit__num__simps_I7_J,axiom,
! [R2: num,M2: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R2 ) @ ( bit1 @ M2 ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M2 ) ) ) ) ).
% take_bit_num_simps(7)
thf(fact_4535_take__bit__num__simps_I6_J,axiom,
! [R2: num,M2: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R2 ) @ ( bit0 @ M2 ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q4: num] : ( some @ num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M2 ) ) ) ).
% take_bit_num_simps(6)
thf(fact_4536_INF__Int__eq,axiom,
! [A: $tType,S3: set @ ( set @ A )] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ ( set @ A ) @ ( A > $o )
@ ^ [I4: set @ A,X4: A] : ( member @ A @ X4 @ I4 )
@ S3 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Inf_Inf @ ( set @ A ) @ S3 ) ) ) ) ).
% INF_Int_eq
thf(fact_4537_SUP__UN__eq,axiom,
! [B: $tType,A: $tType,R2: B > ( set @ A ),S3: set @ B] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I4: B,X4: A] : ( member @ A @ X4 @ ( R2 @ I4 ) )
@ S3 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R2 @ S3 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_4538_Sup__SUP__eq,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( A > $o ) )
= ( ^ [S6: set @ ( A > $o ),X4: A] : ( member @ A @ X4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S6 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_4539_SUP__Sup__eq,axiom,
! [A: $tType,S3: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ ( set @ A ) @ ( A > $o )
@ ^ [I4: set @ A,X4: A] : ( member @ A @ X4 @ I4 )
@ S3 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Sup_Sup @ ( set @ A ) @ S3 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_4540_INF__INT__eq,axiom,
! [B: $tType,A: $tType,R2: B > ( set @ A ),S3: set @ B] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I4: B,X4: A] : ( member @ A @ X4 @ ( R2 @ I4 ) )
@ S3 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R2 @ S3 ) ) ) ) ) ).
% INF_INT_eq
thf(fact_4541_Inf__INT__eq,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( A > $o ) )
= ( ^ [S6: set @ ( A > $o ),X4: A] : ( member @ A @ X4 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S6 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_4542_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I4: C,X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ ( R2 @ I4 ) )
@ S3 ) )
= ( ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_4543_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S3: 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 ),X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ I4 )
@ S3 ) )
= ( ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_4544_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S3: 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 ),X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ I4 )
@ S3 ) )
= ( ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ).
% INF_Int_eq2
thf(fact_4545_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I4: C,X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ ( R2 @ I4 ) )
@ S3 ) )
= ( ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_4546_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S6: set @ ( A > B > $o ),X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ 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 ) @ S6 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_4547_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S6: set @ ( A > B > $o ),X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ 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 ) @ S6 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_4548_and__not__num_Osimps_I1_J,axiom,
( ( bit_and_not_num @ one2 @ one2 )
= ( none @ num ) ) ).
% and_not_num.simps(1)
thf(fact_4549_and__not__num_Osimps_I8_J,axiom,
! [M2: num,N: num] :
( ( bit_and_not_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N10: num] : ( some @ num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M2 @ N ) ) ) ).
% and_not_num.simps(8)
thf(fact_4550_option_Odisc__eq__case_I2_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
= ( case_option @ $o @ A @ $false
@ ^ [Uu3: A] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_4551_option_Odisc__eq__case_I1_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
= ( none @ A ) )
= ( case_option @ $o @ A @ $true
@ ^ [Uu3: A] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_4552_None__notin__image__Some,axiom,
! [A: $tType,A6: set @ A] :
~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A6 ) ) ).
% None_notin_image_Some
thf(fact_4553_INF__filter__not__bot,axiom,
! [I6: $tType,A: $tType,B5: set @ I6,F6: I6 > ( filter @ A )] :
( ! [X10: set @ I6] :
( ( ord_less_eq @ ( set @ I6 ) @ X10 @ B5 )
=> ( ( finite_finite2 @ I6 @ X10 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I6 @ ( filter @ A ) @ F6 @ X10 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I6 @ ( filter @ A ) @ F6 @ B5 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_4554_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A6: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A6 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ( B5 @ X4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_4555_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A6: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ( B5 @ X4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_4556_UN__empty,axiom,
! [B: $tType,A: $tType,B5: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_4557_UN__empty2,axiom,
! [B: $tType,A: $tType,A6: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( bot_bot @ ( set @ A ) )
@ A6 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_4558_UN__subset__iff,axiom,
! [A: $tType,B: $tType,A6: B > ( set @ A ),I5: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I5 ) ) @ B5 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( ord_less_eq @ ( set @ A ) @ ( A6 @ X4 ) @ B5 ) ) ) ) ).
% UN_subset_iff
thf(fact_4559_UN__upper,axiom,
! [B: $tType,A: $tType,A3: A,A6: set @ A,B5: A > ( set @ B )] :
( ( member @ A @ A3 @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ ( B5 @ A3 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A6 ) ) ) ) ).
% UN_upper
thf(fact_4560_UN__least,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: A > ( set @ B ),C4: set @ B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ ( B5 @ X5 ) @ C4 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A6 ) ) @ C4 ) ) ).
% UN_least
thf(fact_4561_UN__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ A,F3: A > ( set @ B ),G3: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ G3 @ B5 ) ) ) ) ) ).
% UN_mono
thf(fact_4562_UN__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A6: set @ A,A3: B,B5: A > ( set @ B )] :
( ( member @ A @ U @ A6 )
=> ( ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( insert2 @ B @ A3 @ ( B5 @ X4 ) )
@ A6 ) )
= ( insert2 @ B @ A3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A6 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_4563_INT__subset__iff,axiom,
! [A: $tType,B: $tType,B5: set @ A,A6: B > ( set @ A ),I5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I5 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ ( A6 @ X4 ) ) ) ) ) ).
% INT_subset_iff
thf(fact_4564_INT__anti__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ A,F3: A > ( set @ B ),G3: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ B5 ) ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ G3 @ A6 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_4565_INT__greatest,axiom,
! [B: $tType,A: $tType,A6: set @ A,C4: set @ B,B5: A > ( set @ B )] :
( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ C4 @ ( B5 @ X5 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ C4 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A6 ) ) ) ) ).
% INT_greatest
thf(fact_4566_INT__lower,axiom,
! [B: $tType,A: $tType,A3: A,A6: set @ A,B5: A > ( set @ B )] :
( ( member @ A @ A3 @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A6 ) ) @ ( B5 @ A3 ) ) ) ).
% INT_lower
thf(fact_4567_INT__extend__simps_I5_J,axiom,
! [I6: $tType,J4: $tType,A3: I6,B5: J4 > ( set @ I6 ),C4: set @ J4] :
( ( insert2 @ I6 @ A3 @ ( complete_Inf_Inf @ ( set @ I6 ) @ ( image2 @ J4 @ ( set @ I6 ) @ B5 @ C4 ) ) )
= ( complete_Inf_Inf @ ( set @ I6 )
@ ( image2 @ J4 @ ( set @ I6 )
@ ^ [X4: J4] : ( insert2 @ I6 @ A3 @ ( B5 @ X4 ) )
@ C4 ) ) ) ).
% INT_extend_simps(5)
thf(fact_4568_INT__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A6: set @ A,A3: B,B5: A > ( set @ B )] :
( ( member @ A @ U @ A6 )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( insert2 @ B @ A3 @ ( B5 @ X4 ) )
@ A6 ) )
= ( insert2 @ B @ A3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A6 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_4569_and__not__num_Osimps_I4_J,axiom,
! [M2: num] :
( ( bit_and_not_num @ ( bit0 @ M2 ) @ one2 )
= ( some @ num @ ( bit0 @ M2 ) ) ) ).
% and_not_num.simps(4)
thf(fact_4570_and__not__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_and_not_num @ one2 @ ( bit0 @ N ) )
= ( some @ num @ one2 ) ) ).
% and_not_num.simps(2)
thf(fact_4571_and__not__num_Osimps_I3_J,axiom,
! [N: num] :
( ( bit_and_not_num @ one2 @ ( bit1 @ N ) )
= ( none @ num ) ) ).
% and_not_num.simps(3)
thf(fact_4572_in__image__insert__iff,axiom,
! [A: $tType,B5: set @ ( set @ A ),X3: A,A6: set @ A] :
( ! [C7: set @ A] :
( ( member @ ( set @ A ) @ C7 @ B5 )
=> ~ ( member @ A @ X3 @ C7 ) )
=> ( ( member @ ( set @ A ) @ A6 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X3 ) @ B5 ) )
= ( ( member @ A @ X3 @ A6 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_4573_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
! [N: nat,M2: num] :
( ( bit_take_bit_num @ N @ ( bit0 @ M2 ) )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N3: nat] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q4: num] : ( some @ num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ N3 @ M2 ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_4574_case__optionE,axiom,
! [A: $tType,P: $o,Q: A > $o,X3: option @ A] :
( ( case_option @ $o @ A @ P @ Q @ X3 )
=> ( ( ( X3
= ( none @ A ) )
=> ~ P )
=> ~ ! [Y4: A] :
( ( X3
= ( some @ A @ Y4 ) )
=> ~ ( Q @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_4575_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C4: set @ B,A3: A,B5: B > ( set @ A )] :
( ( ( C4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ C4 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ C4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert2 @ A @ A3 @ ( B5 @ X4 ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_4576_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C4: set @ C,A6: C > ( set @ D ),B5: set @ D] :
( ( ( C4
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A6 @ C4 ) ) @ B5 )
= B5 ) )
& ( ( C4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A6 @ C4 ) ) @ B5 )
= ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A6 @ X4 ) @ B5 )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_4577_UN__extend__simps_I3_J,axiom,
! [E: $tType,F: $tType,C4: set @ F,A6: set @ E,B5: F > ( set @ E )] :
( ( ( C4
= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A6 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F @ ( set @ E ) @ B5 @ C4 ) ) )
= A6 ) )
& ( ( C4
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A6 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F @ ( set @ E ) @ B5 @ C4 ) ) )
= ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F @ ( set @ E )
@ ^ [X4: F] : ( sup_sup @ ( set @ E ) @ A6 @ ( B5 @ X4 ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_4578_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C4: set @ D,A6: set @ C,B5: D > ( set @ C )] :
( ( ( C4
= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A6 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B5 @ C4 ) ) )
= A6 ) )
& ( ( C4
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A6 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B5 @ C4 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A6 @ ( B5 @ X4 ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_4579_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C4: set @ A,A6: A > ( set @ B ),B5: set @ B] :
( ( ( C4
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ C4 ) ) @ B5 )
= B5 ) )
& ( ( C4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ C4 ) ) @ B5 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A6 @ X4 ) @ B5 )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_4580_bij__betw__UNION__chain,axiom,
! [B: $tType,C: $tType,A: $tType,I5: set @ A,A6: A > ( set @ B ),F3: B > C,A11: A > ( set @ C )] :
( ! [I3: A,J2: A] :
( ( member @ A @ I3 @ I5 )
=> ( ( member @ A @ J2 @ I5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A6 @ I3 ) @ ( A6 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A6 @ J2 ) @ ( A6 @ I3 ) ) ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( bij_betw @ B @ C @ F3 @ ( A6 @ I3 ) @ ( A11 @ I3 ) ) )
=> ( bij_betw @ B @ C @ F3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ I5 ) ) @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ A11 @ I5 ) ) ) ) ) ).
% bij_betw_UNION_chain
thf(fact_4581_Pow__insert,axiom,
! [A: $tType,A3: A,A6: set @ A] :
( ( pow2 @ A @ ( insert2 @ A @ A3 @ A6 ) )
= ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A6 ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ A3 ) @ ( pow2 @ A @ A6 ) ) ) ) ).
% Pow_insert
thf(fact_4582_and__not__num_Osimps_I7_J,axiom,
! [M2: num] :
( ( bit_and_not_num @ ( bit1 @ M2 ) @ one2 )
= ( some @ num @ ( bit0 @ M2 ) ) ) ).
% and_not_num.simps(7)
thf(fact_4583_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X6: set @ A,A6: set @ ( product_prod @ A @ B ),Y8: set @ B,P: A > B > $o,Q: A > B > $o] :
( ( X6
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 ) )
=> ( ( Y8
= ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ Y8 )
=> ( ( P @ X5 @ Xa3 )
=> ( Q @ X5 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_4584_and__not__num__eq__Some__iff,axiom,
! [M2: num,N: num,Q3: num] :
( ( ( bit_and_not_num @ M2 @ N )
= ( some @ num @ Q3 ) )
= ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( numeral_numeral @ int @ Q3 ) ) ) ).
% and_not_num_eq_Some_iff
thf(fact_4585_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
! [N: nat,M2: num] :
( ( bit_take_bit_num @ N @ ( bit1 @ M2 ) )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N3: nat] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N3 @ M2 ) ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_4586_INT__extend__simps_I4_J,axiom,
! [G: $tType,H5: $tType,C4: set @ H5,A6: set @ G,B5: H5 > ( set @ G )] :
( ( ( C4
= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( minus_minus @ ( set @ G ) @ A6 @ ( complete_Sup_Sup @ ( set @ G ) @ ( image2 @ H5 @ ( set @ G ) @ B5 @ C4 ) ) )
= A6 ) )
& ( ( C4
!= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( minus_minus @ ( set @ G ) @ A6 @ ( complete_Sup_Sup @ ( set @ G ) @ ( image2 @ H5 @ ( set @ G ) @ B5 @ C4 ) ) )
= ( complete_Inf_Inf @ ( set @ G )
@ ( image2 @ H5 @ ( set @ G )
@ ^ [X4: H5] : ( minus_minus @ ( set @ G ) @ A6 @ ( B5 @ X4 ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_4587_UN__le__add__shift__strict,axiom,
! [A: $tType,M7: nat > ( set @ A ),K2: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I4: nat] : ( M7 @ ( plus_plus @ nat @ I4 @ K2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M7 @ ( set_or7035219750837199246ssThan @ nat @ K2 @ ( plus_plus @ nat @ N @ K2 ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_4588_UN__le__add__shift,axiom,
! [A: $tType,M7: nat > ( set @ A ),K2: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I4: nat] : ( M7 @ ( plus_plus @ nat @ I4 @ K2 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M7 @ ( set_or1337092689740270186AtMost @ nat @ K2 @ ( plus_plus @ nat @ N @ K2 ) ) ) ) ) ).
% UN_le_add_shift
thf(fact_4589_subset__subseqs,axiom,
! [A: $tType,X6: set @ A,Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ X6 @ ( set2 @ A @ Xs2 ) )
=> ( member @ ( set @ A ) @ X6 @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_4590_subseqs__powset,axiom,
! [A: $tType,Xs2: list @ A] :
( ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) )
= ( pow2 @ A @ ( set2 @ A @ Xs2 ) ) ) ).
% subseqs_powset
thf(fact_4591_sum_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,A6: B > ( set @ C ),G3: C > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I5 )
=> ( finite_finite2 @ C @ ( A6 @ X5 ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I5 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I5 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A6 @ X5 ) @ ( A6 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G3 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A6 @ I5 ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( groups7311177749621191930dd_sum @ C @ A @ G3 @ ( A6 @ X4 ) )
@ I5 ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_4592_prod_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,A6: B > ( set @ C ),G3: C > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I5 )
=> ( finite_finite2 @ C @ ( A6 @ X5 ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I5 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I5 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A6 @ X5 ) @ ( A6 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G3 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A6 @ I5 ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( groups7121269368397514597t_prod @ C @ A @ G3 @ ( A6 @ X4 ) )
@ I5 ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_4593_and__not__num__eq__None__iff,axiom,
! [M2: num,N: num] :
( ( ( bit_and_not_num @ M2 @ N )
= ( none @ num ) )
= ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( zero_zero @ int ) ) ) ).
% and_not_num_eq_None_iff
thf(fact_4594_card__UN__le,axiom,
! [B: $tType,A: $tType,I5: set @ A,A6: A > ( set @ B )] :
( ( finite_finite2 @ A @ I5 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ I5 ) ) )
@ ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] : ( finite_card @ B @ ( A6 @ I4 ) )
@ I5 ) ) ) ).
% card_UN_le
thf(fact_4595_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I5: set @ A,A6: A > ( set @ B )] :
( ( finite_finite2 @ A @ I5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ I5 )
=> ( finite_finite2 @ B @ ( A6 @ X5 ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ I5 )
=> ! [Xa3: A] :
( ( member @ A @ Xa3 @ I5 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ ( A6 @ X5 ) @ ( A6 @ Xa3 ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ I5 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] : ( finite_card @ B @ ( A6 @ I4 ) )
@ I5 ) ) ) ) ) ).
% card_UN_disjoint
thf(fact_4596_int__numeral__and__not__num,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ N ) ) ) ).
% int_numeral_and_not_num
thf(fact_4597_int__numeral__not__and__num,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ N @ M2 ) ) ) ).
% int_numeral_not_and_num
thf(fact_4598_Bit__Operations_Otake__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N3: nat,M5: num] :
( product_case_prod @ nat @ num @ ( option @ num )
@ ^ [A8: nat,X4: num] :
( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [O: nat] :
( case_num @ ( option @ num ) @ ( some @ num @ one2 )
@ ^ [P5: num] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q4: num] : ( some @ num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ O @ P5 ) )
@ ^ [P5: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P5 ) ) )
@ X4 )
@ A8 )
@ ( product_Pair @ nat @ num @ N3 @ M5 ) ) ) ) ).
% Bit_Operations.take_bit_num_code
thf(fact_4599_Union__image__insert,axiom,
! [A: $tType,B: $tType,F3: B > ( set @ A ),A3: B,B5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ ( insert2 @ B @ A3 @ B5 ) ) )
= ( sup_sup @ ( set @ A ) @ ( F3 @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ B5 ) ) ) ) ).
% Union_image_insert
thf(fact_4600_Union__image__empty,axiom,
! [B: $tType,A: $tType,A6: set @ A,F3: B > ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= A6 ) ).
% Union_image_empty
thf(fact_4601_UN__image__subset,axiom,
! [C: $tType,A: $tType,B: $tType,F3: B > ( set @ A ),G3: C > ( set @ B ),X3: C,X6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ ( G3 @ X3 ) ) ) @ X6 )
= ( ord_less_eq @ ( set @ B ) @ ( G3 @ X3 )
@ ( collect @ B
@ ^ [X4: B] : ( ord_less_eq @ ( set @ A ) @ ( F3 @ X4 ) @ X6 ) ) ) ) ).
% UN_image_subset
thf(fact_4602_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I5: set @ A,F6: A > ( filter @ B )] :
( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I5 )
=> ? [X: A] :
( ( member @ A @ X @ I5 )
& ( ord_less_eq @ ( filter @ B ) @ ( F6 @ X ) @ ( inf_inf @ ( filter @ B ) @ ( F6 @ I3 ) @ ( F6 @ J2 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F6 @ I5 ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ I5 )
& ( ( F6 @ X4 )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_4603_Inf__filter__not__bot,axiom,
! [A: $tType,B5: set @ ( filter @ A )] :
( ! [X10: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X10 @ B5 )
=> ( ( finite_finite2 @ ( filter @ A ) @ X10 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ X10 )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ B5 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% Inf_filter_not_bot
thf(fact_4604_verit__eq__simplify_I17_J,axiom,
! [A: $tType,F1: A,F22: num > A,F32: num > A,X2: num] :
( ( case_num @ A @ F1 @ F22 @ F32 @ ( bit0 @ X2 ) )
= ( F22 @ X2 ) ) ).
% verit_eq_simplify(17)
thf(fact_4605_verit__eq__simplify_I16_J,axiom,
! [A: $tType,F1: A,F22: num > A,F32: num > A] :
( ( case_num @ A @ F1 @ F22 @ F32 @ one2 )
= F1 ) ).
% verit_eq_simplify(16)
thf(fact_4606_conj__subset__def,axiom,
! [A: $tType,A6: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A6
@ ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A6 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_4607_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F3: A > C,G3: D > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu3: C] : ( bot_bot @ ( set @ B ) )
@ F3 )
= ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image2 @ D @ B @ G3 )
@ ^ [Uu3: A] : ( bot_bot @ ( set @ D ) ) ) ) ).
% empty_natural
thf(fact_4608_length__remdups__concat,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ ( concat @ A @ Xss ) ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xss ) ) ) ) ) ).
% length_remdups_concat
thf(fact_4609_Pow__fold,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( pow2 @ A @ A6 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A,A7: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A7 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X4 ) @ A7 ) )
@ ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A6 ) ) ) ).
% Pow_fold
thf(fact_4610_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F3: nat > ( set @ A ),S3: set @ A] :
( ! [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( F3 @ I3 ) @ S3 )
=> ( ( finite_finite2 @ A @ S3 )
=> ( ? [N7: nat] :
( ! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ N7 )
=> ! [M: nat] :
( ( ord_less_eq @ nat @ M @ N7 )
=> ( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ ( set @ A ) @ ( F3 @ M ) @ ( F3 @ N2 ) ) ) ) )
& ! [N2: nat] :
( ( ord_less_eq @ nat @ N7 @ N2 )
=> ( ( F3 @ N7 )
= ( F3 @ N2 ) ) ) )
=> ( ( F3 @ ( finite_card @ A @ S3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ F3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_4611_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_4612_UNIV__I,axiom,
! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_4613_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_4614_Int__UNIV,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A6 @ B5 )
= ( top_top @ ( set @ A ) ) )
= ( ( A6
= ( top_top @ ( set @ A ) ) )
& ( B5
= ( top_top @ ( set @ A ) ) ) ) ) ).
% Int_UNIV
thf(fact_4615_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X3: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X3 )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_4616_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X3: A] :
( ( ord_max @ A @ X3 @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_4617_fold__empty,axiom,
! [B: $tType,A: $tType,F3: B > A > A,Z2: A] :
( ( finite_fold @ B @ A @ F3 @ Z2 @ ( bot_bot @ ( set @ B ) ) )
= Z2 ) ).
% fold_empty
thf(fact_4618_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_4619_set__remdups,axiom,
! [A: $tType,Xs2: list @ A] :
( ( set2 @ A @ ( remdups @ A @ Xs2 ) )
= ( set2 @ A @ Xs2 ) ) ).
% set_remdups
thf(fact_4620_length__remdups__eq,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs2 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) )
= ( ( remdups @ A @ Xs2 )
= Xs2 ) ) ).
% length_remdups_eq
thf(fact_4621_Collect__const,axiom,
! [A: $tType,P: $o] :
( ( P
=> ( ( collect @ A
@ ^ [S7: A] : P )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P
=> ( ( collect @ A
@ ^ [S7: A] : P )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_4622_range__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% range_add
thf(fact_4623_surj__plus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_plus
thf(fact_4624_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.compl_zero
thf(fact_4625_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.compl_one
thf(fact_4626_Inf__UNIV,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Inf_UNIV
thf(fact_4627_ccInf__empty,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% ccInf_empty
thf(fact_4628_Inf__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Inf_empty
thf(fact_4629_Diff__UNIV,axiom,
! [A: $tType,A6: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_4630_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_4631_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_4632_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_4633_length__remdups__leq,axiom,
! [A: $tType,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_remdups_leq
thf(fact_4634_range__constant,axiom,
! [B: $tType,A: $tType,X3: A] :
( ( image2 @ B @ A
@ ^ [Uu3: B] : X3
@ ( top_top @ ( set @ B ) ) )
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_4635_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_4636_INT__constant,axiom,
! [B: $tType,A: $tType,A6: set @ B,C3: set @ A] :
( ( ( A6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C3
@ A6 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C3
@ A6 ) )
= C3 ) ) ) ).
% INT_constant
thf(fact_4637_Inf__atMostLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( top_top @ A ) @ X3 )
=> ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X3 ) )
= ( bot_bot @ A ) ) ) ) ).
% Inf_atMostLessThan
thf(fact_4638_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C4: set @ A,A6: A > ( set @ B ),B5: set @ B] :
( ( ( C4
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A6 @ X4 ) @ B5 )
@ C4 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A6 @ X4 ) @ B5 )
@ C4 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ C4 ) ) @ B5 ) ) ) ) ).
% INT_simps(1)
thf(fact_4639_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C4: set @ D,A6: set @ C,B5: D > ( set @ C )] :
( ( ( C4
= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A6 @ ( B5 @ X4 ) )
@ C4 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A6 @ ( B5 @ X4 ) )
@ C4 ) )
= ( inf_inf @ ( set @ C ) @ A6 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B5 @ C4 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_4640_INT__simps_I3_J,axiom,
! [E: $tType,F: $tType,C4: set @ E,A6: E > ( set @ F ),B5: set @ F] :
( ( ( C4
= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image2 @ E @ ( set @ F )
@ ^ [X4: E] : ( minus_minus @ ( set @ F ) @ ( A6 @ X4 ) @ B5 )
@ C4 ) )
= ( top_top @ ( set @ F ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image2 @ E @ ( set @ F )
@ ^ [X4: E] : ( minus_minus @ ( set @ F ) @ ( A6 @ X4 ) @ B5 )
@ C4 ) )
= ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image2 @ E @ ( set @ F ) @ A6 @ C4 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_4641_INT__simps_I4_J,axiom,
! [G: $tType,H5: $tType,C4: set @ H5,A6: set @ G,B5: H5 > ( set @ G )] :
( ( ( C4
= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G )
@ ( image2 @ H5 @ ( set @ G )
@ ^ [X4: H5] : ( minus_minus @ ( set @ G ) @ A6 @ ( B5 @ X4 ) )
@ C4 ) )
= ( top_top @ ( set @ G ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G )
@ ( image2 @ H5 @ ( set @ G )
@ ^ [X4: H5] : ( minus_minus @ ( set @ G ) @ A6 @ ( B5 @ X4 ) )
@ C4 ) )
= ( minus_minus @ ( set @ G ) @ A6 @ ( complete_Sup_Sup @ ( set @ G ) @ ( image2 @ H5 @ ( set @ G ) @ B5 @ C4 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_4642_rangeI,axiom,
! [A: $tType,B: $tType,F3: B > A,X3: B] : ( member @ A @ ( F3 @ X3 ) @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_4643_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A,X3: B] :
( ( B2
= ( F3 @ X3 ) )
=> ( member @ A @ B2 @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_4644_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,G3: B > C] :
( ( image2 @ B @ A
@ ^ [X4: B] : ( F3 @ ( G3 @ X4 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image2 @ C @ A @ F3 @ ( image2 @ B @ C @ G3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_4645_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A] :
( ( member @ A @ B2 @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X5: B] :
( B2
!= ( F3 @ X5 ) ) ) ).
% rangeE
thf(fact_4646_atLeastAtMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( bounded_lattice @ A )
=> ! [X3: A,Y: A] :
( ( ( set_or1337092689740270186AtMost @ A @ X3 @ Y )
= ( top_top @ ( set @ A ) ) )
= ( ( X3
= ( bot_bot @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_4647_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_4648_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_4649_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_4650_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_4651_subset__UNIV,axiom,
! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_4652_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $true ) ) ).
% UNIV_def
thf(fact_4653_Un__UNIV__right,axiom,
! [A: $tType,A6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_4654_Un__UNIV__left,axiom,
! [A: $tType,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B5 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_4655_Int__UNIV__right,axiom,
! [A: $tType,A6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) )
= A6 ) ).
% Int_UNIV_right
thf(fact_4656_Int__UNIV__left,axiom,
! [A: $tType,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B5 )
= B5 ) ).
% Int_UNIV_left
thf(fact_4657_insert__UNIV,axiom,
! [A: $tType,X3: A] :
( ( insert2 @ A @ X3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_4658_less__filter__def,axiom,
! [A: $tType] :
( ( ord_less @ ( filter @ A ) )
= ( ^ [F9: filter @ A,F10: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F9 @ F10 )
& ~ ( ord_less_eq @ ( filter @ A ) @ F10 @ F9 ) ) ) ) ).
% less_filter_def
thf(fact_4659_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_4660_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A3 ) ) ).
% top.extremum_strict
thf(fact_4661_UNIV__witness,axiom,
! [A: $tType] :
? [X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_4662_UNIV__eq__I,axiom,
! [A: $tType,A6: set @ A] :
( ! [X5: A] : ( member @ A @ X5 @ A6 )
=> ( ( top_top @ ( set @ A ) )
= A6 ) ) ).
% UNIV_eq_I
thf(fact_4663_UNIV__option__conv,axiom,
! [A: $tType] :
( ( top_top @ ( set @ ( option @ A ) ) )
= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% UNIV_option_conv
thf(fact_4664_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_4665_range__subsetD,axiom,
! [B: $tType,A: $tType,F3: B > A,B5: set @ A,I: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) @ B5 )
=> ( member @ A @ ( F3 @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_4666_perfect__space__class_OUNIV__not__singleton,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [X3: A] :
( ( top_top @ ( set @ A ) )
!= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% perfect_space_class.UNIV_not_singleton
thf(fact_4667_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_4668_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_4669_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_4670_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_4671_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_4672_Compl__partition,axiom,
! [A: $tType,A6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition
thf(fact_4673_Compl__partition2,axiom,
! [A: $tType,A6: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ A6 )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition2
thf(fact_4674_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_4675_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_4676_sup__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ( sup_sup @ A @ X3 @ Y )
= ( top_top @ A ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) ) ) ).
% sup_shunt
thf(fact_4677_boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [A3: A,X3: A,Y: A] :
( ( ( inf_inf @ A @ A3 @ X3 )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A3 @ X3 )
= ( top_top @ A ) )
=> ( ( ( inf_inf @ A @ A3 @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A3 @ Y )
= ( top_top @ A ) )
=> ( X3 = Y ) ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_4678_union__fold__insert,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( sup_sup @ ( set @ A ) @ A6 @ B5 )
= ( finite_fold @ A @ ( set @ A ) @ ( insert2 @ A ) @ B5 @ A6 ) ) ) ).
% union_fold_insert
thf(fact_4679_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: A,X3: B] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( F3 @ X3 )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_4680_INF__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,C3: A] :
( ( ( A6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C3
@ A6 ) )
= ( top_top @ A ) ) )
& ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C3
@ A6 ) )
= C3 ) ) ) ) ).
% INF_constant
thf(fact_4681_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_4682_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_4683_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_4684_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: 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 @ A6 ) ) @ ( image2 @ B @ A @ F3 @ ( uminus_uminus @ ( set @ B ) @ A6 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_4685_length__remdups__card__conv,axiom,
! [A: $tType,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs2 ) )
= ( finite_card @ A @ ( set2 @ A @ Xs2 ) ) ) ).
% length_remdups_card_conv
thf(fact_4686_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_4687_notin__range__Some,axiom,
! [A: $tType,X3: option @ A] :
( ( ~ ( member @ ( option @ A ) @ X3 @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
= ( X3
= ( none @ A ) ) ) ).
% notin_range_Some
thf(fact_4688_INT__empty,axiom,
! [B: $tType,A: $tType,B5: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_4689_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ( inf_inf @ A @ X3 @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ X3 @ Y )
= ( top_top @ A ) )
=> ( ( uminus_uminus @ A @ X3 )
= Y ) ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_4690_Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( complete_Sup_Sup @ A @ A6 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) @ A6 ) ) ) ) ).
% Sup_fold_sup
thf(fact_4691_sum_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups7311177749621191930dd_sum @ B @ A )
= ( ^ [G4: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% sum.eq_fold
thf(fact_4692_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_4693_image__fold__insert,axiom,
! [B: $tType,A: $tType,A6: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( image2 @ A @ B @ F3 @ A6 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K3: A] : ( insert2 @ B @ ( F3 @ K3 ) )
@ ( bot_bot @ ( set @ B ) )
@ A6 ) ) ) ).
% image_fold_insert
thf(fact_4694_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_4695_INT__extend__simps_I3_J,axiom,
! [F: $tType,E: $tType,C4: set @ E,A6: E > ( set @ F ),B5: set @ F] :
( ( ( C4
= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image2 @ E @ ( set @ F ) @ A6 @ C4 ) ) @ B5 )
= ( minus_minus @ ( set @ F ) @ ( top_top @ ( set @ F ) ) @ B5 ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image2 @ E @ ( set @ F ) @ A6 @ C4 ) ) @ B5 )
= ( complete_Inf_Inf @ ( set @ F )
@ ( image2 @ E @ ( set @ F )
@ ^ [X4: E] : ( minus_minus @ ( set @ F ) @ ( A6 @ X4 ) @ B5 )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_4696_UN__finite__subset,axiom,
! [A: $tType,A6: nat > ( set @ A ),C4: set @ A] :
( ! [N2: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) @ C4 ) ) ).
% UN_finite_subset
thf(fact_4697_UN__finite2__eq,axiom,
! [A: $tType,A6: nat > ( set @ A ),B5: nat > ( set @ A ),K2: nat] :
( ! [N2: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N2 @ K2 ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B5 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_4698_SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F3 ) @ ( bot_bot @ A ) @ A6 ) ) ) ) ).
% SUP_fold_sup
thf(fact_4699_UN__finite2__subset,axiom,
! [A: $tType,A6: nat > ( set @ A ),B5: nat > ( set @ A ),K2: nat] :
( ! [N2: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N2 @ K2 ) ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B5 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_4700_Set__filter__fold,axiom,
! [A: $tType,A6: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A6 )
=> ( ( filter3 @ A @ P @ A6 )
= ( finite_fold @ A @ ( set @ A )
@ ^ [X4: A,A16: set @ A] : ( if @ ( set @ A ) @ ( P @ X4 ) @ ( insert2 @ A @ X4 @ A16 ) @ A16 )
@ ( bot_bot @ ( set @ A ) )
@ A6 ) ) ) ).
% Set_filter_fold
thf(fact_4701_image__Fpow__mono,axiom,
! [B: $tType,A: $tType,F3: B > A,A6: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A6 ) @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F3 ) @ ( finite_Fpow @ B @ A6 ) ) @ ( finite_Fpow @ A @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_4702_and__not__num_Oelims,axiom,
! [X3: num,Xa2: num,Y: option @ num] :
( ( ( bit_and_not_num @ X3 @ Xa2 )
= Y )
=> ( ( ( X3 = one2 )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ( ( X3 = one2 )
=> ( ? [N2: num] :
( Xa2
= ( bit0 @ N2 ) )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X3 = one2 )
=> ( ? [N2: num] :
( Xa2
= ( bit1 @ N2 ) )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( some @ num @ ( bit0 @ M ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit0 @ N2 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit1 @ N2 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( some @ num @ ( bit0 @ M ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit0 @ N2 ) )
=> ( Y
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N10: num] : ( some @ num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M @ N2 ) ) ) ) )
=> ~ ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit1 @ N2 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
thf(fact_4703_member__filter,axiom,
! [A: $tType,X3: A,P: A > $o,A6: set @ A] :
( ( member @ A @ X3 @ ( filter3 @ A @ P @ A6 ) )
= ( ( member @ A @ X3 @ A6 )
& ( P @ X3 ) ) ) ).
% member_filter
thf(fact_4704_map__option__eq__Some,axiom,
! [A: $tType,B: $tType,F3: B > A,Xo: option @ B,Y: A] :
( ( ( map_option @ B @ A @ F3 @ Xo )
= ( some @ A @ Y ) )
= ( ? [Z4: B] :
( ( Xo
= ( some @ B @ Z4 ) )
& ( ( F3 @ Z4 )
= Y ) ) ) ) ).
% map_option_eq_Some
thf(fact_4705_option_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: option @ A] :
( ( ( map_option @ A @ B @ F3 @ A3 )
= ( none @ B ) )
= ( A3
= ( none @ A ) ) ) ).
% option.map_disc_iff
thf(fact_4706_map__option__is__None,axiom,
! [A: $tType,B: $tType,F3: B > A,Opt: option @ B] :
( ( ( map_option @ B @ A @ F3 @ Opt )
= ( none @ A ) )
= ( Opt
= ( none @ B ) ) ) ).
% map_option_is_None
thf(fact_4707_None__eq__map__option__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,X3: option @ B] :
( ( ( none @ A )
= ( map_option @ B @ A @ F3 @ X3 ) )
= ( X3
= ( none @ B ) ) ) ).
% None_eq_map_option_iff
thf(fact_4708_card__UNIV__bool,axiom,
( ( finite_card @ $o @ ( top_top @ ( set @ $o ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% card_UNIV_bool
thf(fact_4709_case__map__option,axiom,
! [B: $tType,A: $tType,C: $tType,G3: A,H: B > A,F3: C > B,X3: option @ C] :
( ( case_option @ A @ B @ G3 @ H @ ( map_option @ C @ B @ F3 @ X3 ) )
= ( case_option @ A @ C @ G3 @ ( comp @ B @ A @ C @ H @ F3 ) @ X3 ) ) ).
% case_map_option
thf(fact_4710_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P: $o] :
( ( P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A8: A,B8: B] : P ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A8: A,B8: B] : P ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_4711_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_4712_map__option_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F3: B > C,G3: A > B] :
( ( comp @ ( option @ B ) @ ( option @ C ) @ ( option @ A ) @ ( map_option @ B @ C @ F3 ) @ ( map_option @ A @ B @ G3 ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ F3 @ G3 ) ) ) ).
% map_option.comp
thf(fact_4713_option_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G3: B > C,F3: A > B,V2: option @ A] :
( ( map_option @ B @ C @ G3 @ ( map_option @ A @ B @ F3 @ V2 ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ G3 @ F3 ) @ V2 ) ) ).
% option.map_comp
thf(fact_4714_map__option_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F3: B > C,G3: A > B,Option: option @ A] :
( ( map_option @ B @ C @ F3 @ ( map_option @ A @ B @ G3 @ Option ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ F3 @ G3 ) @ Option ) ) ).
% map_option.compositionality
thf(fact_4715_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_4716_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_4717_option_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F3: A > B,X2: A] :
( ( map_option @ A @ B @ F3 @ ( some @ A @ X2 ) )
= ( some @ B @ ( F3 @ X2 ) ) ) ).
% option.simps(9)
thf(fact_4718_map__option__cong,axiom,
! [B: $tType,A: $tType,X3: option @ A,Y: option @ A,F3: A > B,G3: A > B] :
( ( X3 = Y )
=> ( ! [A5: A] :
( ( Y
= ( some @ A @ A5 ) )
=> ( ( F3 @ A5 )
= ( G3 @ A5 ) ) )
=> ( ( map_option @ A @ B @ F3 @ X3 )
= ( map_option @ A @ B @ G3 @ Y ) ) ) ) ).
% map_option_cong
thf(fact_4719_option_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F3: A > B] :
( ( map_option @ A @ B @ F3 @ ( none @ A ) )
= ( none @ B ) ) ).
% option.simps(8)
thf(fact_4720_Set_Ofilter__def,axiom,
! [A: $tType] :
( ( filter3 @ A )
= ( ^ [P4: A > $o,A7: set @ A] :
( collect @ A
@ ^ [A8: A] :
( ( member @ A @ A8 @ A7 )
& ( P4 @ A8 ) ) ) ) ) ).
% Set.filter_def
thf(fact_4721_option_Omap__ident,axiom,
! [A: $tType,T2: option @ A] :
( ( map_option @ A @ A
@ ^ [X4: A] : X4
@ T2 )
= T2 ) ).
% option.map_ident
thf(fact_4722_and__not__num_Osimps_I5_J,axiom,
! [M2: num,N: num] :
( ( bit_and_not_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M2 @ N ) ) ) ).
% and_not_num.simps(5)
thf(fact_4723_empty__in__Fpow,axiom,
! [A: $tType,A6: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( finite_Fpow @ A @ A6 ) ) ).
% empty_in_Fpow
thf(fact_4724_option_Omap__sel,axiom,
! [B: $tType,A: $tType,A3: option @ A,F3: A > B] :
( ( A3
!= ( none @ A ) )
=> ( ( the2 @ B @ ( map_option @ A @ B @ F3 @ A3 ) )
= ( F3 @ ( the2 @ A @ A3 ) ) ) ) ).
% option.map_sel
thf(fact_4725_Inter__UNIV,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Inter_UNIV
thf(fact_4726_and__not__num_Osimps_I9_J,axiom,
! [M2: num,N: num] :
( ( bit_and_not_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M2 @ N ) ) ) ).
% and_not_num.simps(9)
thf(fact_4727_and__not__num_Osimps_I6_J,axiom,
! [M2: num,N: num] :
( ( bit_and_not_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M2 @ N ) ) ) ).
% and_not_num.simps(6)
thf(fact_4728_option_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F3: B > nat,G3: A > B] :
( ( comp @ ( option @ B ) @ nat @ ( option @ A ) @ ( size_option @ B @ F3 ) @ ( map_option @ A @ B @ G3 ) )
= ( size_option @ A @ ( comp @ B @ nat @ A @ F3 @ G3 ) ) ) ).
% option.size_gen_o_map
thf(fact_4729_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu3: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_4730_map__option__case,axiom,
! [A: $tType,B: $tType] :
( ( map_option @ B @ A )
= ( ^ [F4: B > A] :
( case_option @ ( option @ A ) @ B @ ( none @ A )
@ ^ [X4: B] : ( some @ A @ ( F4 @ X4 ) ) ) ) ) ).
% map_option_case
thf(fact_4731_Fpow__mono,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A6 ) @ ( finite_Fpow @ A @ B5 ) ) ) ).
% Fpow_mono
thf(fact_4732_Fpow__def,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A7: set @ A] :
( collect @ ( set @ A )
@ ^ [X8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ X8 @ A7 )
& ( finite_finite2 @ A @ X8 ) ) ) ) ) ).
% Fpow_def
thf(fact_4733_fold__union__pair,axiom,
! [B: $tType,A: $tType,B5: set @ A,X3: B,A6: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ B5 )
=> ( ( 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 @ X3 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B5 ) )
@ A6 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) )
@ A6
@ B5 ) ) ) ).
% fold_union_pair
thf(fact_4734_map__option__o__empty,axiom,
! [C: $tType,B: $tType,A: $tType,F3: C > B] :
( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F3 )
@ ^ [X4: A] : ( none @ C ) )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% map_option_o_empty
thf(fact_4735_root__def,axiom,
( root
= ( ^ [N3: nat,X4: real] :
( if @ real
@ ( N3
= ( 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 ) @ N3 ) )
@ X4 ) ) ) ) ).
% root_def
thf(fact_4736_and__num_Oelims,axiom,
! [X3: num,Xa2: num,Y: option @ num] :
( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 )
= Y )
=> ( ( ( X3 = one2 )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X3 = one2 )
=> ( ? [N2: num] :
( Xa2
= ( bit0 @ N2 ) )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ( ( X3 = one2 )
=> ( ? [N2: num] :
( Xa2
= ( bit1 @ N2 ) )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ? [M: num] :
( X3
= ( bit0 @ M ) )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit0 @ N2 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit1 @ N2 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ) )
=> ( ( ? [M: num] :
( X3
= ( bit1 @ M ) )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit0 @ N2 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ) )
=> ~ ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit1 @ N2 ) )
=> ( Y
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N10: num] : ( some @ num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
thf(fact_4737_top1I,axiom,
! [A: $tType,X3: A] : ( top_top @ ( A > $o ) @ X3 ) ).
% top1I
thf(fact_4738_top2I,axiom,
! [A: $tType,B: $tType,X3: A,Y: B] : ( top_top @ ( A > B > $o ) @ X3 @ Y ) ).
% top2I
thf(fact_4739_UNIV__bool,axiom,
( ( top_top @ ( set @ $o ) )
= ( insert2 @ $o @ $false @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% UNIV_bool
thf(fact_4740_and__num_Osimps_I1_J,axiom,
( ( bit_un7362597486090784418nd_num @ one2 @ one2 )
= ( some @ num @ one2 ) ) ).
% and_num.simps(1)
thf(fact_4741_and__num_Osimps_I5_J,axiom,
! [M2: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ).
% and_num.simps(5)
thf(fact_4742_and__num_Osimps_I7_J,axiom,
! [M2: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M2 ) @ one2 )
= ( some @ num @ one2 ) ) ).
% and_num.simps(7)
thf(fact_4743_and__num_Osimps_I3_J,axiom,
! [N: num] :
( ( bit_un7362597486090784418nd_num @ one2 @ ( bit1 @ N ) )
= ( some @ num @ one2 ) ) ).
% and_num.simps(3)
thf(fact_4744_and__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_un7362597486090784418nd_num @ one2 @ ( bit0 @ N ) )
= ( none @ num ) ) ).
% and_num.simps(2)
thf(fact_4745_and__num_Osimps_I4_J,axiom,
! [M2: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M2 ) @ one2 )
= ( none @ num ) ) ).
% and_num.simps(4)
thf(fact_4746_and__num__eq__Some__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: num,Q3: num] :
( ( ( bit_un7362597486090784418nd_num @ M2 @ N )
= ( some @ num @ Q3 ) )
= ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ Q3 ) ) ) ) ).
% and_num_eq_Some_iff
thf(fact_4747_and__num_Osimps_I6_J,axiom,
! [M2: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ).
% and_num.simps(6)
thf(fact_4748_and__num_Osimps_I8_J,axiom,
! [M2: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ).
% and_num.simps(8)
thf(fact_4749_and__num__eq__None__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: num] :
( ( ( bit_un7362597486090784418nd_num @ M2 @ N )
= ( none @ num ) )
= ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% and_num_eq_None_iff
thf(fact_4750_numeral__and__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ) ).
% numeral_and_num
thf(fact_4751_and__num_Osimps_I9_J,axiom,
! [M2: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N10: num] : ( some @ num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ).
% and_num.simps(9)
thf(fact_4752_card__UNIV__char,axiom,
( ( finite_card @ char @ ( top_top @ ( set @ char ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% card_UNIV_char
thf(fact_4753_xor__num_Oelims,axiom,
! [X3: num,Xa2: num,Y: option @ num] :
( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 )
= Y )
=> ( ( ( X3 = one2 )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa2
= ( bit0 @ N2 ) )
=> ( Y
!= ( some @ num @ ( bit1 @ N2 ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa2
= ( bit1 @ N2 ) )
=> ( Y
!= ( some @ num @ ( bit0 @ N2 ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( some @ num @ ( bit1 @ M ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit0 @ N2 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit1 @ N2 ) )
=> ( Y
!= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ( ( Xa2 = one2 )
=> ( Y
!= ( some @ num @ ( bit0 @ M ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit0 @ N2 ) )
=> ( Y
!= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ) )
=> ~ ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit1 @ N2 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.elims
thf(fact_4754_these__insert__Some,axiom,
! [A: $tType,X3: A,A6: set @ ( option @ A )] :
( ( these @ A @ ( insert2 @ ( option @ A ) @ ( some @ A @ X3 ) @ A6 ) )
= ( insert2 @ A @ X3 @ ( these @ A @ A6 ) ) ) ).
% these_insert_Some
thf(fact_4755_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_4756_these__image__Some__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( these @ A @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A6 ) )
= A6 ) ).
% these_image_Some_eq
thf(fact_4757_these__insert__None,axiom,
! [A: $tType,A6: set @ ( option @ A )] :
( ( these @ A @ ( insert2 @ ( option @ A ) @ ( none @ A ) @ A6 ) )
= ( these @ A @ A6 ) ) ).
% these_insert_None
thf(fact_4758_in__these__eq,axiom,
! [A: $tType,X3: A,A6: set @ ( option @ A )] :
( ( member @ A @ X3 @ ( these @ A @ A6 ) )
= ( member @ ( option @ A ) @ ( some @ A @ X3 ) @ A6 ) ) ).
% in_these_eq
thf(fact_4759_xor__num_Osimps_I1_J,axiom,
( ( bit_un2480387367778600638or_num @ one2 @ one2 )
= ( none @ num ) ) ).
% xor_num.simps(1)
thf(fact_4760_xor__num_Osimps_I5_J,axiom,
! [M2: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ).
% xor_num.simps(5)
thf(fact_4761_xor__num__eq__Some__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: num,Q3: num] :
( ( ( bit_un2480387367778600638or_num @ M2 @ N )
= ( some @ num @ Q3 ) )
= ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ Q3 ) ) ) ) ).
% xor_num_eq_Some_iff
thf(fact_4762_xor__num_Osimps_I9_J,axiom,
! [M2: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ).
% xor_num.simps(9)
thf(fact_4763_Option_Othese__def,axiom,
! [A: $tType] :
( ( these @ A )
= ( ^ [A7: set @ ( option @ A )] :
( image2 @ ( option @ A ) @ A @ ( the2 @ A )
@ ( collect @ ( option @ A )
@ ^ [X4: option @ A] :
( ( member @ ( option @ A ) @ X4 @ A7 )
& ( X4
!= ( none @ A ) ) ) ) ) ) ) ).
% Option.these_def
thf(fact_4764_xor__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_un2480387367778600638or_num @ one2 @ ( bit0 @ N ) )
= ( some @ num @ ( bit1 @ N ) ) ) ).
% xor_num.simps(2)
thf(fact_4765_xor__num_Osimps_I3_J,axiom,
! [N: num] :
( ( bit_un2480387367778600638or_num @ one2 @ ( bit1 @ N ) )
= ( some @ num @ ( bit0 @ N ) ) ) ).
% xor_num.simps(3)
thf(fact_4766_xor__num_Osimps_I4_J,axiom,
! [M2: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M2 ) @ one2 )
= ( some @ num @ ( bit1 @ M2 ) ) ) ).
% xor_num.simps(4)
thf(fact_4767_xor__num_Osimps_I7_J,axiom,
! [M2: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M2 ) @ one2 )
= ( some @ num @ ( bit0 @ M2 ) ) ) ).
% xor_num.simps(7)
thf(fact_4768_xor__num__eq__None__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: num] :
( ( ( bit_un2480387367778600638or_num @ M2 @ N )
= ( none @ num ) )
= ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% xor_num_eq_None_iff
thf(fact_4769_numeral__xor__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).
% numeral_xor_num
thf(fact_4770_these__empty__eq,axiom,
! [A: $tType,B5: set @ ( option @ A )] :
( ( ( these @ A @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B5
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B5
= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_4771_these__not__empty__eq,axiom,
! [A: $tType,B5: set @ ( option @ A )] :
( ( ( these @ A @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B5
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B5
!= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_4772_Some__image__these__eq,axiom,
! [A: $tType,A6: set @ ( option @ A )] :
( ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A6 ) )
= ( collect @ ( option @ A )
@ ^ [X4: option @ A] :
( ( member @ ( option @ A ) @ X4 @ A6 )
& ( X4
!= ( none @ A ) ) ) ) ) ).
% Some_image_these_eq
thf(fact_4773_xor__num_Osimps_I6_J,axiom,
! [M2: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).
% xor_num.simps(6)
thf(fact_4774_xor__num_Osimps_I8_J,axiom,
! [M2: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).
% xor_num.simps(8)
thf(fact_4775_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_4776_char__of__quasi__inj,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: A,N: A] :
( ( ( unique5772411509450598832har_of @ A @ M2 )
= ( unique5772411509450598832har_of @ A @ N ) )
= ( ( modulo_modulo @ A @ M2 @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) )
= ( modulo_modulo @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% char_of_quasi_inj
thf(fact_4777_char__of__mod__256,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: A] :
( ( unique5772411509450598832har_of @ A @ ( modulo_modulo @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
= ( unique5772411509450598832har_of @ A @ N ) ) ) ).
% char_of_mod_256
thf(fact_4778_char__of__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,M2: A] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N )
=> ( ( unique5772411509450598832har_of @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ M2 ) )
= ( unique5772411509450598832har_of @ A @ M2 ) ) ) ) ).
% char_of_take_bit_eq
thf(fact_4779_of__char__of,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [A3: A] :
( ( comm_s6883823935334413003f_char @ A @ ( unique5772411509450598832har_of @ A @ A3 ) )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% of_char_of
thf(fact_4780_char__of__def,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( unique5772411509450598832har_of @ A )
= ( ^ [N3: A] :
( char2
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( one_one @ nat ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% char_of_def
thf(fact_4781_of__char__mod__256,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [C3: char] :
( ( modulo_modulo @ A @ ( comm_s6883823935334413003f_char @ A @ C3 ) @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) )
= ( comm_s6883823935334413003f_char @ A @ C3 ) ) ) ).
% of_char_mod_256
thf(fact_4782_char_Osize_I2_J,axiom,
! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_size @ char @ ( char2 @ X1 @ X2 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size(2)
thf(fact_4783_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_4784_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_4785_char__of__eq__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: A,C3: char] :
( ( ( unique5772411509450598832har_of @ A @ N )
= C3 )
= ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N )
= ( comm_s6883823935334413003f_char @ A @ C3 ) ) ) ) ).
% char_of_eq_iff
thf(fact_4786_integer__of__char__code,axiom,
! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
= ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( zero_neq_one_of_bool @ code_integer @ B72 ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B62 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B52 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B42 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B32 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B22 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B1 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B0 ) ) ) ).
% integer_of_char_code
thf(fact_4787_of__char__Char,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( comm_s6883823935334413003f_char @ A @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
= ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cons @ $o @ B0 @ ( cons @ $o @ B1 @ ( cons @ $o @ B22 @ ( cons @ $o @ B32 @ ( cons @ $o @ B42 @ ( cons @ $o @ B52 @ ( cons @ $o @ B62 @ ( cons @ $o @ B72 @ ( nil @ $o ) ) ) ) ) ) ) ) ) ) ) ) ).
% of_char_Char
thf(fact_4788_Id__on__fold,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( id_on @ A @ A6 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X4: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A6 ) ) ) ).
% Id_on_fold
thf(fact_4789_Id__onI,axiom,
! [A: $tType,A3: A,A6: set @ A] :
( ( member @ A @ A3 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id_on @ A @ A6 ) ) ) ).
% Id_onI
thf(fact_4790_list_Osimps_I15_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( set2 @ A @ ( cons @ A @ X21 @ X222 ) )
= ( insert2 @ A @ X21 @ ( set2 @ A @ X222 ) ) ) ).
% list.simps(15)
thf(fact_4791_nth__Cons__Suc,axiom,
! [A: $tType,X3: A,Xs2: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X3 @ Xs2 ) @ ( suc @ N ) )
= ( nth @ A @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_4792_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_4793_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F3: B > A,A3: A,X3: B,Xs2: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F3 @ A3 @ ( cons @ B @ X3 @ Xs2 ) )
= ( plus_plus @ A @ ( F3 @ X3 ) @ ( times_times @ A @ A3 @ ( groups4207007520872428315er_sum @ B @ A @ F3 @ A3 @ Xs2 ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_4794_enumerate__simps_I2_J,axiom,
! [B: $tType,N: nat,X3: B,Xs2: list @ B] :
( ( enumerate @ B @ N @ ( cons @ B @ X3 @ Xs2 ) )
= ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N @ X3 ) @ ( enumerate @ B @ ( suc @ N ) @ Xs2 ) ) ) ).
% enumerate_simps(2)
thf(fact_4795_nth__Cons__numeral,axiom,
! [A: $tType,X3: A,Xs2: list @ A,V2: num] :
( ( nth @ A @ ( cons @ A @ X3 @ Xs2 ) @ ( numeral_numeral @ nat @ V2 ) )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_4796_Id__onE,axiom,
! [A: $tType,C3: product_prod @ A @ A,A6: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A6 ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( C3
!= ( product_Pair @ A @ A @ X5 @ X5 ) ) ) ) ).
% Id_onE
thf(fact_4797_Id__on__eqI,axiom,
! [A: $tType,A3: A,B2: A,A6: set @ A] :
( ( A3 = B2 )
=> ( ( member @ A @ A3 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id_on @ A @ A6 ) ) ) ) ).
% Id_on_eqI
thf(fact_4798_Id__on__iff,axiom,
! [A: $tType,X3: A,Y: A,A6: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( id_on @ A @ A6 ) )
= ( ( X3 = Y )
& ( member @ A @ X3 @ A6 ) ) ) ).
% Id_on_iff
thf(fact_4799_splice_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ~ ! [X5: A,Xs3: list @ A,Ys4: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_4800_shuffles_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( ! [Xs3: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ ( nil @ A ) ) )
=> ~ ! [X5: A,Xs3: list @ A,Y4: A,Ys4: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ A @ Y4 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_4801_sorted__wrt_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P7: A > A > $o] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( nil @ A ) ) )
=> ~ ! [P7: A > A > $o,X5: A,Ys4: list @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X5 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_4802_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X3: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F2: A > B,X5: A] :
( X3
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F2 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) )
=> ( ! [F2: A > B,X5: A,Y4: A,Zs: list @ A] :
( X3
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F2 @ ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Zs ) ) ) )
=> ~ ! [A5: A > B] :
( X3
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A5 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_4803_successively_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P7: A > A > $o] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( nil @ A ) ) )
=> ( ! [P7: A > A > $o,X5: A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) )
=> ~ ! [P7: A > A > $o,X5: A,Y4: A,Xs3: list @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_4804_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X3: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F2: A > B,Bs2: list @ B] :
( X3
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs2 ) ) )
=> ~ ! [F2: A > B,A5: A,As: list @ A,Bs2: list @ B] :
( X3
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_4805_set__ConsD,axiom,
! [A: $tType,Y: A,X3: A,Xs2: list @ A] :
( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X3 @ Xs2 ) ) )
=> ( ( Y = X3 )
| ( member @ A @ Y @ ( set2 @ A @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_4806_list_Oset__cases,axiom,
! [A: $tType,E3: A,A3: list @ A] :
( ( member @ A @ E3 @ ( set2 @ A @ A3 ) )
=> ( ! [Z22: list @ A] :
( A3
!= ( cons @ A @ E3 @ Z22 ) )
=> ~ ! [Z1: A,Z22: list @ A] :
( ( A3
= ( cons @ A @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E3 @ ( set2 @ A @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_4807_list_Oset__intros_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] : ( member @ A @ X21 @ ( set2 @ A @ ( cons @ A @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_4808_list_Oset__intros_I2_J,axiom,
! [A: $tType,Y: A,X222: list @ A,X21: A] :
( ( member @ A @ Y @ ( set2 @ A @ X222 ) )
=> ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_4809_length__Suc__conv,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N ) )
= ( ? [Y3: A,Ys3: list @ A] :
( ( Xs2
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_4810_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( suc @ N )
= ( size_size @ ( list @ A ) @ Xs2 ) )
= ( ? [Y3: A,Ys3: list @ A] :
( ( Xs2
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_4811_length__Cons,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_Cons
thf(fact_4812_set__subset__Cons,axiom,
! [A: $tType,Xs2: list @ A,X3: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ ( cons @ A @ X3 @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_4813_impossible__Cons,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,X3: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs2
!= ( cons @ A @ X3 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_4814_list__induct2,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X5: A,Xs3: list @ A,Y4: B,Ys4: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_4815_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs2: list @ A,Ys: list @ B,Zs2: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X5: A,Xs3: list @ A,Y4: B,Ys4: list @ B,Z3: C,Zs: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs )
=> ( P @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) @ ( cons @ C @ Z3 @ Zs ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_4816_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Xs2: list @ A,Ys: list @ B,Zs2: list @ C,Ws: list @ D,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs2 )
= ( size_size @ ( list @ D ) @ Ws ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
=> ( ! [X5: A,Xs3: list @ A,Y4: B,Ys4: list @ B,Z3: C,Zs: list @ C,W2: D,Ws2: list @ D] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs )
= ( size_size @ ( list @ D ) @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs @ Ws2 )
=> ( P @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) @ ( cons @ C @ Z3 @ Zs ) @ ( cons @ D @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_4817_Id__on__def_H,axiom,
! [A: $tType,A6: A > $o] :
( ( id_on @ A @ ( collect @ A @ A6 ) )
= ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y3: A] :
( ( X4 = Y3 )
& ( A6 @ X4 ) ) ) ) ) ).
% Id_on_def'
thf(fact_4818_distinct_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( distinct @ A @ ( cons @ A @ X3 @ Xs2 ) )
= ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
& ( distinct @ A @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_4819_list__update__code_I3_J,axiom,
! [A: $tType,X3: A,Xs2: list @ A,I: nat,Y: A] :
( ( list_update @ A @ ( cons @ A @ X3 @ Xs2 ) @ ( suc @ I ) @ Y )
= ( cons @ A @ X3 @ ( list_update @ A @ Xs2 @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_4820_replicate__Suc,axiom,
! [A: $tType,N: nat,X3: A] :
( ( replicate @ A @ ( suc @ N ) @ X3 )
= ( cons @ A @ X3 @ ( replicate @ A @ N @ X3 ) ) ) ).
% replicate_Suc
thf(fact_4821_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Y: B,Ys: list @ B] :
( ( ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ X3 @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ X3 @ ( cons @ B @ Y @ Ys ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ X3 @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ Y @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Ys ) ) ) ) ) ) ).
% insort_key.simps(2)
thf(fact_4822_remdups_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( remdups @ A @ ( cons @ A @ X3 @ Xs2 ) )
= ( remdups @ A @ Xs2 ) ) )
& ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( remdups @ A @ ( cons @ A @ X3 @ Xs2 ) )
= ( cons @ A @ X3 @ ( remdups @ A @ Xs2 ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_4823_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= ( ? [X4: A,Ys3: list @ A] :
( ( Xs2
= ( cons @ A @ X4 @ Ys3 ) )
& ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_4824_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ B,F3: B > A,A3: B] :
( ! [X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ Xs2 ) )
=> ( ord_less_eq @ A @ ( F3 @ A3 ) @ ( F3 @ X5 ) ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ A3 @ Xs2 )
= ( cons @ B @ A3 @ Xs2 ) ) ) ) ).
% insort_is_Cons
thf(fact_4825_count__list_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Y: A,Xs2: list @ A] :
( ( ( X3 = Y )
=> ( ( count_list @ A @ ( cons @ A @ X3 @ Xs2 ) @ Y )
= ( plus_plus @ nat @ ( count_list @ A @ Xs2 @ Y ) @ ( one_one @ nat ) ) ) )
& ( ( X3 != Y )
=> ( ( count_list @ A @ ( cons @ A @ X3 @ Xs2 ) @ Y )
= ( count_list @ A @ Xs2 @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_4826_the__elem__set,axiom,
! [A: $tType,X3: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= X3 ) ).
% the_elem_set
thf(fact_4827_list_Osize_I4_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X222 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size(4)
thf(fact_4828_list_Osize__gen_I2_J,axiom,
! [A: $tType,X3: A > nat,X21: A,X222: list @ A] :
( ( size_list @ A @ X3 @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X3 @ X21 ) @ ( size_list @ A @ X3 @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_4829_nth__equal__first__eq,axiom,
! [A: $tType,X3: A,Xs2: list @ A,N: nat] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( nth @ A @ ( cons @ A @ X3 @ Xs2 ) @ N )
= X3 )
= ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_4830_Pow__set_I2_J,axiom,
! [B: $tType,X3: B,Xs2: list @ B] :
( ( pow2 @ B @ ( set2 @ B @ ( cons @ B @ X3 @ Xs2 ) ) )
= ( sup_sup @ ( set @ ( set @ B ) ) @ ( pow2 @ B @ ( set2 @ B @ Xs2 ) ) @ ( image2 @ ( set @ B ) @ ( set @ B ) @ ( insert2 @ B @ X3 ) @ ( pow2 @ B @ ( set2 @ B @ Xs2 ) ) ) ) ) ).
% Pow_set(2)
thf(fact_4831_Id__on__def,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A7: set @ A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X4: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A7 ) ) ) ) ).
% Id_on_def
thf(fact_4832_String_Ochar__of__ascii__of,axiom,
! [C3: char] :
( ( comm_s6883823935334413003f_char @ nat @ ( ascii_of @ C3 ) )
= ( bit_se2584673776208193580ke_bit @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( comm_s6883823935334413003f_char @ nat @ C3 ) ) ) ).
% String.char_of_ascii_of
thf(fact_4833_concat__inth,axiom,
! [A: $tType,Xs2: list @ A,X3: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs2 @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= X3 ) ).
% concat_inth
thf(fact_4834_DERIV__even__real__root,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X3 @ ( 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 @ X3 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_4835_append__eq__append__conv,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,Us: list @ A,Vs: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys ) )
| ( ( size_size @ ( list @ A ) @ Us )
= ( size_size @ ( list @ A ) @ Vs ) ) )
=> ( ( ( append @ A @ Xs2 @ Us )
= ( append @ A @ Ys @ Vs ) )
= ( ( Xs2 = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_4836_sorted__list__of__set__lessThan__Suc,axiom,
! [K2: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ ( suc @ K2 ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ K2 ) ) @ ( cons @ nat @ K2 @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_4837_sorted__list__of__set__atMost__Suc,axiom,
! [K2: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ ( suc @ K2 ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ K2 ) ) @ ( cons @ nat @ ( suc @ K2 ) @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_4838_length__append,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% length_append
thf(fact_4839_set__append,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( append @ A @ Xs2 @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ Ys ) ) ) ).
% set_append
thf(fact_4840_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_4841_size__list__append,axiom,
! [A: $tType,F3: A > nat,Xs2: list @ A,Ys: list @ A] :
( ( size_list @ A @ F3 @ ( append @ A @ Xs2 @ Ys ) )
= ( plus_plus @ nat @ ( size_list @ A @ F3 @ Xs2 ) @ ( size_list @ A @ F3 @ Ys ) ) ) ).
% size_list_append
thf(fact_4842_nth__append__length,axiom,
! [A: $tType,Xs2: list @ A,X3: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= X3 ) ).
% nth_append_length
thf(fact_4843_nth__append__length__plus,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,N: nat] :
( ( nth @ A @ ( append @ A @ Xs2 @ Ys ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) )
= ( nth @ A @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_4844_list__update__length,axiom,
! [A: $tType,Xs2: list @ A,X3: A,Ys: list @ A,Y: A] :
( ( list_update @ A @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) @ Y )
= ( append @ A @ Xs2 @ ( cons @ A @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_4845_distinct__append,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( append @ A @ Xs2 @ Ys ) )
= ( ( distinct @ A @ Xs2 )
& ( distinct @ A @ Ys )
& ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% distinct_append
thf(fact_4846_enumerate__append__eq,axiom,
! [A: $tType,N: nat,Xs2: list @ A,Ys: list @ A] :
( ( enumerate @ A @ N @ ( append @ A @ Xs2 @ Ys ) )
= ( append @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs2 ) @ ( enumerate @ A @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_4847_DERIV__at__within__shift__lemma,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Y: A,Z2: A,X3: A,S3: set @ A] :
( ( has_field_derivative @ A @ F3 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z2 @ X3 ) @ ( image2 @ A @ A @ ( plus_plus @ A @ Z2 ) @ S3 ) ) )
=> ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F3 @ ( plus_plus @ A @ Z2 ) ) @ Y @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ).
% DERIV_at_within_shift_lemma
thf(fact_4848_DERIV__at__within__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Y: A,Z2: A,X3: A,S3: set @ A] :
( ( has_field_derivative @ A @ F3 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z2 @ X3 ) @ ( image2 @ A @ A @ ( plus_plus @ A @ Z2 ) @ S3 ) ) )
= ( has_field_derivative @ A
@ ^ [X4: A] : ( F3 @ ( plus_plus @ A @ Z2 @ X4 ) )
@ Y
@ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ).
% DERIV_at_within_shift
thf(fact_4849_DERIV__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,F8: A,X3: A,S: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_field_derivative @ A @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ) ).
% DERIV_subset
thf(fact_4850_has__field__derivative__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Y: A,X3: A,S: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F3 @ Y @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_field_derivative @ A @ F3 @ Y @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ) ).
% has_field_derivative_subset
thf(fact_4851_field__differentiable__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,F8: A,F6: filter @ A,G3: A > A,G6: A] :
( ( has_field_derivative @ A @ F3 @ F8 @ F6 )
=> ( ( has_field_derivative @ A @ G3 @ G6 @ F6 )
=> ( has_field_derivative @ A
@ ^ [Z4: A] : ( plus_plus @ A @ ( F3 @ Z4 ) @ ( G3 @ Z4 ) )
@ ( plus_plus @ A @ F8 @ G6 )
@ F6 ) ) ) ) ).
% field_differentiable_add
thf(fact_4852_DERIV__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Da: A,X3: A,S: set @ A,G3: A > A,Db: A] :
( ( has_field_derivative @ A @ F3 @ Da @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( has_field_derivative @ A @ G3 @ Db @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( plus_plus @ A @ ( times_times @ A @ Da @ ( G3 @ X3 ) ) @ ( times_times @ A @ Db @ ( F3 @ X3 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% DERIV_mult
thf(fact_4853_DERIV__mult_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X3: A,S: set @ A,G3: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( has_field_derivative @ A @ G3 @ E5 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( F3 @ X3 ) @ E5 ) @ ( times_times @ A @ D4 @ ( G3 @ X3 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% DERIV_mult'
thf(fact_4854_DERIV__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X3: A,S: set @ A,G3: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( has_field_derivative @ A @ G3 @ E5 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( plus_plus @ A @ D4 @ E5 )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% DERIV_add
thf(fact_4855_DERIV__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Y: A,X3: A,Z2: A] :
( ( has_field_derivative @ A @ F3 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ X3 @ Z2 ) @ ( top_top @ ( set @ A ) ) ) )
= ( has_field_derivative @ A
@ ^ [X4: A] : ( F3 @ ( plus_plus @ A @ X4 @ Z2 ) )
@ Y
@ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_shift
thf(fact_4856_split__list,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ? [Ys4: list @ A,Zs: list @ A] :
( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs ) ) ) ) ).
% split_list
thf(fact_4857_split__list__last,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ? [Ys4: list @ A,Zs: list @ A] :
( ( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs ) ) )
& ~ ( member @ A @ X3 @ ( set2 @ A @ Zs ) ) ) ) ).
% split_list_last
thf(fact_4858_split__list__prop,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
& ( P @ X ) )
=> ? [Ys4: list @ A,X5: A] :
( ? [Zs: list @ A] :
( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) )
& ( P @ X5 ) ) ) ).
% split_list_prop
thf(fact_4859_split__list__first,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ? [Ys4: list @ A,Zs: list @ A] :
( ( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs ) ) )
& ~ ( member @ A @ X3 @ ( set2 @ A @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_4860_split__list__propE,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
& ( P @ X ) )
=> ~ ! [Ys4: list @ A,X5: A] :
( ? [Zs: list @ A] :
( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) )
=> ~ ( P @ X5 ) ) ) ).
% split_list_propE
thf(fact_4861_append__Cons__eq__iff,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Ys: list @ A,Xs4: list @ A,Ys5: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ~ ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
=> ( ( ( append @ A @ Xs2 @ ( cons @ A @ X3 @ Ys ) )
= ( append @ A @ Xs4 @ ( cons @ A @ X3 @ Ys5 ) ) )
= ( ( Xs2 = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_4862_in__set__conv__decomp,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_4863_split__list__last__prop,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
& ( P @ X ) )
=> ? [Ys4: list @ A,X5: A,Zs: list @ A] :
( ( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) )
& ( P @ X5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Zs ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_4864_split__list__first__prop,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
& ( P @ X ) )
=> ? [Ys4: list @ A,X5: A] :
( ? [Zs: list @ A] :
( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) )
& ( P @ X5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_4865_split__list__last__propE,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
& ( P @ X ) )
=> ~ ! [Ys4: list @ A,X5: A,Zs: list @ A] :
( ( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) )
=> ( ( P @ X5 )
=> ~ ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Zs ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_4866_split__list__first__propE,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
& ( P @ X ) )
=> ~ ! [Ys4: list @ A,X5: A] :
( ? [Zs: list @ A] :
( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) )
=> ( ( P @ X5 )
=> ~ ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_4867_in__set__conv__decomp__last,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs3 ) ) )
& ~ ( member @ A @ X3 @ ( set2 @ A @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_4868_in__set__conv__decomp__first,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs3 ) ) )
& ~ ( member @ A @ X3 @ ( set2 @ A @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_4869_split__list__last__prop__iff,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list @ A,X4: A,Zs3: list @ A] :
( ( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_4870_split__list__first__prop__iff,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list @ A,X4: A] :
( ? [Zs3: list @ A] :
( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_4871_replicate__add,axiom,
! [A: $tType,N: nat,M2: nat,X3: A] :
( ( replicate @ A @ ( plus_plus @ nat @ N @ M2 ) @ X3 )
= ( append @ A @ ( replicate @ A @ N @ X3 ) @ ( replicate @ A @ M2 @ X3 ) ) ) ).
% replicate_add
thf(fact_4872_remove1__append,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Ys: list @ A] :
( ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( remove1 @ A @ X3 @ ( append @ A @ Xs2 @ Ys ) )
= ( append @ A @ ( remove1 @ A @ X3 @ Xs2 ) @ Ys ) ) )
& ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( remove1 @ A @ X3 @ ( append @ A @ Xs2 @ Ys ) )
= ( append @ A @ Xs2 @ ( remove1 @ A @ X3 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_4873_at__le,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,T2: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
=> ( ord_less_eq @ ( filter @ A ) @ ( topolo174197925503356063within @ A @ X3 @ S ) @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ).
% at_le
thf(fact_4874_DERIV__cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K2: A,Xa2: A] :
( has_field_derivative @ A
@ ^ [X4: A] : ( cos @ A @ ( plus_plus @ A @ X4 @ K2 ) )
@ ( uminus_uminus @ A @ ( sin @ A @ ( plus_plus @ A @ Xa2 @ K2 ) ) )
@ ( topolo174197925503356063within @ A @ Xa2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% DERIV_cos_add
thf(fact_4875_DERIV__power__Suc,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X3: A,S: set @ A,N: nat] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( power_power @ A @ ( F3 @ X4 ) @ ( suc @ N ) )
@ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) @ ( times_times @ A @ D4 @ ( power_power @ A @ ( F3 @ X3 ) @ N ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ).
% DERIV_power_Suc
thf(fact_4876_DERIV__const__average,axiom,
! [A3: real,B2: real,V2: real > real,K2: real] :
( ( A3 != B2 )
=> ( ! [X5: real] : ( has_field_derivative @ real @ V2 @ K2 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( V2 @ ( divide_divide @ real @ ( plus_plus @ real @ A3 @ B2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( divide_divide @ real @ ( plus_plus @ real @ ( V2 @ A3 ) @ ( V2 @ B2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% DERIV_const_average
thf(fact_4877_DERIV__inverse,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X3: A,S: set @ A] :
( ( X3
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( inverse_inverse @ A ) @ ( uminus_uminus @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ).
% DERIV_inverse
thf(fact_4878_DERIV__power,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X3: A,S: set @ A,N: nat] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( power_power @ A @ ( F3 @ X4 ) @ N )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( times_times @ A @ D4 @ ( power_power @ A @ ( F3 @ X3 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ).
% DERIV_power
thf(fact_4879_same__length__different,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( Xs2 != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ? [Pre: list @ A,X5: A,Xs5: list @ A,Y4: A,Ys6: list @ A] :
( ( X5 != Y4 )
& ( Xs2
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X5 @ ( nil @ A ) ) @ Xs5 ) ) )
& ( Ys
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_4880_DERIV__pow,axiom,
! [N: nat,X3: real,S: set @ real] :
( has_field_derivative @ real
@ ^ [X4: real] : ( power_power @ real @ X4 @ N )
@ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ X3 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ S ) ) ).
% DERIV_pow
thf(fact_4881_termdiffs__strong__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,X3: A] :
( ! [Y4: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C3 @ N3 ) @ ( power_power @ A @ Y4 @ N3 ) ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C3 @ N3 ) @ ( power_power @ A @ X4 @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% termdiffs_strong_converges_everywhere
thf(fact_4882_not__distinct__conv__prefix,axiom,
! [A: $tType,As2: list @ A] :
( ( ~ ( distinct @ A @ As2 ) )
= ( ? [Xs: list @ A,Y3: A,Ys3: list @ A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Xs ) )
& ( distinct @ A @ Xs )
& ( As2
= ( append @ A @ Xs @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_4883_DERIV__fun__pow,axiom,
! [G3: real > real,M2: real,X3: real,N: nat] :
( ( has_field_derivative @ real @ G3 @ M2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X4: real] : ( power_power @ real @ ( G3 @ X4 ) @ N )
@ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( G3 @ X3 ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) @ M2 )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_fun_pow
thf(fact_4884_list__update__append1,axiom,
! [A: $tType,I: nat,Xs2: list @ A,Ys: list @ A,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ I @ X3 )
= ( append @ A @ ( list_update @ A @ Xs2 @ I @ X3 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_4885_remove1__split,axiom,
! [A: $tType,A3: A,Xs2: list @ A,Ys: list @ A] :
( ( member @ A @ A3 @ ( set2 @ A @ Xs2 ) )
=> ( ( ( remove1 @ A @ A3 @ Xs2 )
= Ys )
= ( ? [Ls: list @ A,Rs: list @ A] :
( ( Xs2
= ( append @ A @ Ls @ ( cons @ A @ A3 @ Rs ) ) )
& ~ ( member @ A @ A3 @ ( set2 @ A @ Ls ) )
& ( Ys
= ( append @ A @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_4886_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_4887_DERIV__quotient,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X3: A,S: set @ A,G3: A > A,E3: A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( has_field_derivative @ A @ G3 @ E3 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( G3 @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [Y3: A] : ( divide_divide @ A @ ( F3 @ Y3 ) @ ( G3 @ Y3 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D3 @ ( G3 @ X3 ) ) @ ( times_times @ A @ E3 @ ( F3 @ X3 ) ) ) @ ( power_power @ A @ ( G3 @ X3 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ).
% DERIV_quotient
thf(fact_4888_DERIV__inverse__fun,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X3: A,S: set @ A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( F3 @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( inverse_inverse @ A @ ( F3 @ X4 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ D3 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F3 @ X3 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% DERIV_inverse_fun
thf(fact_4889_termdiffs__sums__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K5: real,C3: nat > A,F3: A > A,F8: A,Z2: A] :
( ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C3 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) )
@ ( F3 @ Z3 ) ) )
=> ( ( has_field_derivative @ A @ F3 @ F8 @ ( topolo174197925503356063within @ A @ Z2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K5 )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N3 ) @ ( power_power @ A @ Z2 @ N3 ) )
@ F8 ) ) ) ) ) ).
% termdiffs_sums_strong
thf(fact_4890_length__append__singleton,axiom,
! [A: $tType,Xs2: list @ A,X3: A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_append_singleton
thf(fact_4891_length__Suc__conv__rev,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N ) )
= ( ? [Y3: A,Ys3: list @ A] :
( ( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_4892_termdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K5: A,X3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C3 @ N3 ) @ ( power_power @ A @ K5 @ N3 ) ) )
=> ( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N3 ) @ ( power_power @ A @ K5 @ N3 ) ) )
=> ( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C3 ) @ N3 ) @ ( power_power @ A @ K5 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C3 @ N3 ) @ ( power_power @ A @ X4 @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% termdiffs
thf(fact_4893_termdiffs__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K5: A,X3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C3 @ N3 ) @ ( power_power @ A @ K5 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C3 @ N3 ) @ ( power_power @ A @ X4 @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong
thf(fact_4894_termdiffs__strong_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K5: real,C3: nat > A,Z2: A] :
( ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C3 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K5 )
=> ( has_field_derivative @ A
@ ^ [Z4: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C3 @ N3 ) @ ( power_power @ A @ Z4 @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N3 ) @ ( power_power @ A @ Z2 @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ Z2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong'
thf(fact_4895_nth__append,axiom,
! [A: $tType,N: nat,Xs2: list @ A,Ys: list @ A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( append @ A @ Xs2 @ Ys ) @ N )
= ( nth @ A @ Xs2 @ N ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( append @ A @ Xs2 @ Ys ) @ N )
= ( nth @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_4896_list__update__append,axiom,
! [A: $tType,N: nat,Xs2: list @ A,Ys: list @ A,X3: A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ N @ X3 )
= ( append @ A @ ( list_update @ A @ Xs2 @ N @ X3 ) @ Ys ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ N @ X3 )
= ( append @ A @ Xs2 @ ( list_update @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ X3 ) ) ) ) ) ).
% list_update_append
thf(fact_4897_DERIV__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( tan @ A ) @ ( inverse_inverse @ A @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_tan
thf(fact_4898_artanh__real__has__field__derivative,axiom,
! [X3: real,A6: set @ real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ ( artanh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ A6 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_4899_DERIV__real__sqrt,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( has_field_derivative @ real @ sqrt @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_real_sqrt
thf(fact_4900_DERIV__arctan,axiom,
! [X3: real] : ( has_field_derivative @ real @ arctan @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ).
% DERIV_arctan
thf(fact_4901_arsinh__real__has__field__derivative,axiom,
! [X3: real,A6: set @ real] : ( has_field_derivative @ real @ ( arsinh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ A6 ) ) ).
% arsinh_real_has_field_derivative
thf(fact_4902_DERIV__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( sin @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( cot @ A ) @ ( uminus_uminus @ A @ ( inverse_inverse @ A @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_cot
thf(fact_4903_has__field__derivative__tanh,axiom,
! [A17: $tType] :
( ( ( real_Vector_banach @ A17 )
& ( real_V3459762299906320749_field @ A17 ) )
=> ! [G3: A17 > A17,X3: A17,Db: A17,S: set @ A17] :
( ( ( cosh @ A17 @ ( G3 @ X3 ) )
!= ( zero_zero @ A17 ) )
=> ( ( has_field_derivative @ A17 @ G3 @ Db @ ( topolo174197925503356063within @ A17 @ X3 @ S ) )
=> ( has_field_derivative @ A17
@ ^ [X4: A17] : ( tanh @ A17 @ ( G3 @ X4 ) )
@ ( times_times @ A17 @ ( minus_minus @ A17 @ ( one_one @ A17 ) @ ( power_power @ A17 @ ( tanh @ A17 @ ( G3 @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db )
@ ( topolo174197925503356063within @ A17 @ X3 @ S ) ) ) ) ) ).
% has_field_derivative_tanh
thf(fact_4904_DERIV__real__sqrt__generic,axiom,
! [X3: real,D4: real] :
( ( X3
!= ( zero_zero @ real ) )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( D4
= ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( D4
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( has_field_derivative @ real @ sqrt @ D4 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_4905_arcosh__real__has__field__derivative,axiom,
! [X3: real,A6: set @ real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( has_field_derivative @ real @ ( arcosh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ A6 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_4906_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F3: B > A,A3: A,Xs2: list @ B,Ys: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F3 @ A3 @ ( append @ B @ Xs2 @ Ys ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F3 @ A3 @ Xs2 ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( size_size @ ( list @ B ) @ Xs2 ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F3 @ A3 @ Ys ) ) ) ) ) ).
% horner_sum_append
thf(fact_4907_DERIV__real__root,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X3 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_real_root
thf(fact_4908_DERIV__arccos,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arccos @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arccos
thf(fact_4909_DERIV__arcsin,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arcsin @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arcsin
thf(fact_4910_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F3: real > real,X3: real,N: nat] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T6: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T6 ) @ ( abs_abs @ real @ X3 ) )
& ( ( F3 @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X3 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_4911_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F3: real > real,X3: real,N: nat] :
( ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
& ! [M: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T6: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T6 ) @ ( abs_abs @ real @ X3 ) )
& ( ( F3 @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X3 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X3 @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_4912_DERIV__odd__real__root,axiom,
! [N: nat,X3: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( X3
!= ( 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 @ X3 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_4913_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 )
=> ( ! [M: nat,T6: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less_eq @ real @ T6 @ H ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ 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
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_4914_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 )
=> ( ! [M: nat,T6: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less_eq @ real @ T6 @ H ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ 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
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_4915_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 )
=> ( ! [M: nat,T6: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ H @ T6 )
& ( ord_less_eq @ real @ T6 @ ( zero_zero @ real ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ 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
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_4916_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F3: real > real,N: nat,X3: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X3
!= ( zero_zero @ real ) )
=> ( ! [M: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( 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 @ X3 ) )
& ( ( F3 @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X3 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X3 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_4917_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F3: real > real,N: nat,X3: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M: nat,T6: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ ( abs_abs @ real @ T6 ) @ ( abs_abs @ real @ X3 ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo174197925503356063within @ real @ T6 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T6: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T6 ) @ ( abs_abs @ real @ X3 ) )
& ( ( F3 @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X3 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T6 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_4918_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A3: real,B2: real,C3: real,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M: nat,T6: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ A3 @ T6 )
& ( ord_less_eq @ real @ T6 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo174197925503356063within @ real @ T6 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A3 @ C3 )
=> ( ( ord_less_eq @ real @ C3 @ B2 )
=> ( ( ord_less_eq @ real @ A3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ( ( X3 != C3 )
=> ? [T6: real] :
( ( ( ord_less @ real @ X3 @ C3 )
=> ( ( ord_less @ real @ X3 @ T6 )
& ( ord_less @ real @ T6 @ C3 ) ) )
& ( ~ ( ord_less @ real @ X3 @ C3 )
=> ( ( ord_less @ real @ C3 @ T6 )
& ( ord_less @ real @ T6 @ X3 ) ) )
& ( ( F3 @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ C3 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X3 @ C3 ) @ M5 ) )
@ ( 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 @ X3 @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_4919_Taylor__up,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A3: real,B2: real,C3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M: nat,T6: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ A3 @ T6 )
& ( ord_less_eq @ real @ T6 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo174197925503356063within @ real @ T6 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A3 @ C3 )
=> ( ( ord_less @ real @ C3 @ B2 )
=> ? [T6: real] :
( ( ord_less @ real @ C3 @ T6 )
& ( ord_less @ real @ T6 @ B2 )
& ( ( F3 @ B2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ C3 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C3 ) @ M5 ) )
@ ( 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 @ B2 @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_4920_Taylor__down,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A3: real,B2: real,C3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M: nat,T6: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ A3 @ T6 )
& ( ord_less_eq @ real @ T6 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo174197925503356063within @ real @ T6 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less @ real @ A3 @ C3 )
=> ( ( ord_less_eq @ real @ C3 @ B2 )
=> ? [T6: real] :
( ( ord_less @ real @ A3 @ T6 )
& ( ord_less @ real @ T6 @ C3 )
& ( ( F3 @ A3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ C3 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A3 @ C3 ) @ M5 ) )
@ ( 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_4921_Maclaurin__lemma2,axiom,
! [N: nat,H: real,Diff: nat > real > real,K2: nat,B5: real] :
( ! [M: nat,T6: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T6 )
& ( ord_less_eq @ real @ T6 @ H ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo174197925503356063within @ real @ T6 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( N
= ( suc @ K2 ) )
=> ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ 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 @ B5 @ ( divide_divide @ real @ ( power_power @ real @ U2 @ ( minus_minus @ nat @ N @ M3 ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ M3 ) ) ) ) ) )
@ ( minus_minus @ real @ ( Diff @ ( suc @ M3 ) @ T7 )
@ ( 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 @ T7 @ P5 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) )
@ ( times_times @ real @ B5 @ ( divide_divide @ real @ ( power_power @ real @ T7 @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_4922_DERIV__arctan__series,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real
@ ^ [X9: real] :
( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X9 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) )
@ ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( power_power @ real @ X3 @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_arctan_series
thf(fact_4923_DERIV__real__root__generic,axiom,
! [N: nat,X3: real,D4: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X3
!= ( zero_zero @ real ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( D4
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X3 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( D4
= ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X3 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( D4
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X3 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ D4 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_4924_DERIV__power__series_H,axiom,
! [R: real,F3: nat > real,X0: real] :
( ! [X5: real] :
( ( member @ real @ X5 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( summable @ real
@ ^ [N3: nat] : ( times_times @ real @ ( times_times @ real @ ( F3 @ N3 ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) @ ( power_power @ real @ X5 @ N3 ) ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( has_field_derivative @ real
@ ^ [X4: real] :
( suminf @ real
@ ^ [N3: nat] : ( times_times @ real @ ( F3 @ N3 ) @ ( power_power @ real @ X4 @ ( suc @ N3 ) ) ) )
@ ( suminf @ real
@ ^ [N3: nat] : ( times_times @ real @ ( times_times @ real @ ( F3 @ N3 ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) @ ( power_power @ real @ X0 @ N3 ) ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_4925_has__derivative__arcsin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X3: A,G6: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G3 @ X3 ) )
=> ( ( ord_less @ real @ ( G3 @ X3 ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( arcsin @ ( G3 @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G6 @ X4 ) @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G3 @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ).
% has_derivative_arcsin
thf(fact_4926_has__derivative__arccos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X3: A,G6: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G3 @ X3 ) )
=> ( ( ord_less @ real @ ( G3 @ X3 ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( arccos @ ( G3 @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G6 @ X4 ) @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G3 @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ).
% has_derivative_arccos
thf(fact_4927_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K2: A] :
( ( ord_less_eq @ A @ L @ K2 )
=> ( ( set_or5935395276787703475ssThan @ A @ K2 @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanLessThan_empty
thf(fact_4928_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_4929_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_4930_list__encode_Ocases,axiom,
! [X3: list @ nat] :
( ( X3
!= ( nil @ nat ) )
=> ~ ! [X5: nat,Xs3: list @ nat] :
( X3
!= ( cons @ nat @ X5 @ Xs3 ) ) ) ).
% list_encode.cases
thf(fact_4931_has__derivative__scaleR,axiom,
! [C: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ D )
& ( real_V822414075346904944vector @ C ) )
=> ! [F3: D > real,F8: D > real,X3: D,S: set @ D,G3: D > C,G6: D > C] :
( ( has_derivative @ D @ real @ F3 @ F8 @ ( topolo174197925503356063within @ D @ X3 @ S ) )
=> ( ( has_derivative @ D @ C @ G3 @ G6 @ ( topolo174197925503356063within @ D @ X3 @ S ) )
=> ( has_derivative @ D @ C
@ ^ [X4: D] : ( real_V8093663219630862766scaleR @ C @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ^ [H2: D] : ( plus_plus @ C @ ( real_V8093663219630862766scaleR @ C @ ( F3 @ X3 ) @ ( G6 @ H2 ) ) @ ( real_V8093663219630862766scaleR @ C @ ( F8 @ H2 ) @ ( G3 @ X3 ) ) )
@ ( topolo174197925503356063within @ D @ X3 @ S ) ) ) ) ) ).
% has_derivative_scaleR
thf(fact_4932_has__derivative__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F8: A > B,F6: filter @ A,G3: A > B,G6: A > B] :
( ( has_derivative @ A @ B @ F3 @ F8 @ F6 )
=> ( ( has_derivative @ A @ B @ G3 @ G6 @ F6 )
=> ( has_derivative @ A @ B
@ ^ [X4: A] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ^ [X4: A] : ( plus_plus @ B @ ( F8 @ X4 ) @ ( G6 @ X4 ) )
@ F6 ) ) ) ) ).
% has_derivative_add
thf(fact_4933_has__derivative__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F8: A > B,X3: A,S: set @ A,T2: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ) ).
% has_derivative_subset
thf(fact_4934_has__derivative__mult,axiom,
! [A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ D )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F3: D > A,F8: D > A,X3: D,S: set @ D,G3: D > A,G6: D > A] :
( ( has_derivative @ D @ A @ F3 @ F8 @ ( topolo174197925503356063within @ D @ X3 @ S ) )
=> ( ( has_derivative @ D @ A @ G3 @ G6 @ ( topolo174197925503356063within @ D @ X3 @ S ) )
=> ( has_derivative @ D @ A
@ ^ [X4: D] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ^ [H2: D] : ( plus_plus @ A @ ( times_times @ A @ ( F3 @ X3 ) @ ( G6 @ H2 ) ) @ ( times_times @ A @ ( F8 @ H2 ) @ ( G3 @ X3 ) ) )
@ ( topolo174197925503356063within @ D @ X3 @ S ) ) ) ) ) ).
% has_derivative_mult
thf(fact_4935_has__derivative__in__compose2,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [T2: set @ A,G3: A > B,G6: A > A > B,F3: C > A,S: set @ C,X3: C,F8: C > A] :
( ! [X5: A] :
( ( member @ A @ X5 @ T2 )
=> ( has_derivative @ A @ B @ G3 @ ( G6 @ X5 ) @ ( topolo174197925503356063within @ A @ X5 @ T2 ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ F3 @ S ) @ T2 )
=> ( ( member @ C @ X3 @ S )
=> ( ( has_derivative @ C @ A @ F3 @ F8 @ ( topolo174197925503356063within @ C @ X3 @ S ) )
=> ( has_derivative @ C @ B
@ ^ [X4: C] : ( G3 @ ( F3 @ X4 ) )
@ ^ [Y3: C] : ( G6 @ ( F3 @ X3 ) @ ( F8 @ Y3 ) )
@ ( topolo174197925503356063within @ C @ X3 @ S ) ) ) ) ) ) ) ).
% has_derivative_in_compose2
thf(fact_4936_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C3 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_4937_ivl__disj__int__two_I5_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(5)
thf(fact_4938_ivl__disj__int__two_I4_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(4)
thf(fact_4939_ivl__disj__int__two_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(1)
thf(fact_4940_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_4941_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_4942_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_4943_ivl__disj__un__two_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_4944_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_4945_ivl__disj__un__two_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_4946_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_4947_has__derivative__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,F8: A > B,X3: A,S3: set @ A,N: nat] :
( ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X3 @ S3 ) )
=> ( has_derivative @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N )
@ ^ [Y3: A] : ( times_times @ B @ ( times_times @ B @ ( semiring_1_of_nat @ B @ N ) @ ( F8 @ Y3 ) ) @ ( power_power @ B @ ( F3 @ X3 ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ).
% has_derivative_power
thf(fact_4948_has__derivative__divide,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: C > A,F8: C > A,X3: C,S3: set @ C,G3: C > A,G6: C > A] :
( ( has_derivative @ C @ A @ F3 @ F8 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) )
=> ( ( has_derivative @ C @ A @ G3 @ G6 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) )
=> ( ( ( G3 @ X3 )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ^ [H2: C] : ( plus_plus @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( F3 @ X3 ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G3 @ X3 ) ) @ ( G6 @ H2 ) ) @ ( inverse_inverse @ A @ ( G3 @ X3 ) ) ) ) @ ( divide_divide @ A @ ( F8 @ H2 ) @ ( G3 @ X3 ) ) )
@ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) ) ) ) ) ).
% has_derivative_divide
thf(fact_4949_has__derivative__prod,axiom,
! [B: $tType,I6: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [I5: set @ I6,F3: I6 > A > B,F8: I6 > A > B,X3: A,S3: set @ A] :
( ! [I3: I6] :
( ( member @ I6 @ I3 @ I5 )
=> ( has_derivative @ A @ B @ ( F3 @ I3 ) @ ( F8 @ I3 ) @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) )
=> ( has_derivative @ A @ B
@ ^ [X4: A] :
( groups7121269368397514597t_prod @ I6 @ B
@ ^ [I4: I6] : ( F3 @ I4 @ X4 )
@ I5 )
@ ^ [Y3: A] :
( groups7311177749621191930dd_sum @ I6 @ B
@ ^ [I4: I6] :
( times_times @ B @ ( F8 @ I4 @ Y3 )
@ ( groups7121269368397514597t_prod @ I6 @ B
@ ^ [J3: I6] : ( F3 @ J3 @ X3 )
@ ( minus_minus @ ( set @ I6 ) @ I5 @ ( insert2 @ I6 @ I4 @ ( bot_bot @ ( set @ I6 ) ) ) ) ) )
@ I5 )
@ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ).
% has_derivative_prod
thf(fact_4950_has__derivative__real__sqrt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X3: A,G6: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X3 ) )
=> ( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( sqrt @ ( G3 @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G6 @ X4 ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G3 @ X3 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% has_derivative_real_sqrt
thf(fact_4951_has__derivative__arctan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,G6: A > real,X3: A,S: set @ A] :
( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( arctan @ ( G3 @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G6 @ X4 ) @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G3 @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ).
% has_derivative_arctan
thf(fact_4952_has__derivative__tan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X3: A,G6: A > real,S: set @ A] :
( ( ( cos @ real @ ( G3 @ X3 ) )
!= ( zero_zero @ real ) )
=> ( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( tan @ real @ ( G3 @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G6 @ X4 ) @ ( inverse_inverse @ real @ ( power_power @ real @ ( cos @ real @ ( G3 @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% has_derivative_tan
thf(fact_4953_termdiffs__aux,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K5: A,X3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C3 ) @ N3 ) @ ( power_power @ A @ K5 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( filterlim @ A @ A
@ ^ [H2: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C3 @ N3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X3 @ H2 ) @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ X3 @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_aux
thf(fact_4954_extract__SomeE,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A,Ys: list @ A,Y: A,Zs2: list @ A] :
( ( ( extract @ A @ P @ Xs2 )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs2 ) ) ) )
=> ( ( Xs2
= ( append @ A @ Ys @ ( cons @ A @ Y @ Zs2 ) ) )
& ( P @ Y )
& ~ ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Ys ) )
& ( P @ X ) ) ) ) ).
% extract_SomeE
thf(fact_4955_extract__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A,Ys: list @ A,Y: A,Zs2: list @ A] :
( ( ( extract @ A @ P @ Xs2 )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs2 ) ) ) )
= ( ( Xs2
= ( append @ A @ Ys @ ( cons @ A @ Y @ Zs2 ) ) )
& ( P @ Y )
& ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
& ( P @ X4 ) ) ) ) ).
% extract_Some_iff
thf(fact_4956_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ ( suc @ L ) ) ) ).
% card_greaterThanLessThan
thf(fact_4957_power__tendsto__0__iff,axiom,
! [A: $tType,N: nat,F3: A > real,F6: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real
@ ^ [X4: A] : ( power_power @ real @ ( F3 @ X4 ) @ N )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F6 )
= ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F6 ) ) ) ).
% power_tendsto_0_iff
thf(fact_4958_filterlim__mono,axiom,
! [B: $tType,A: $tType,F3: A > B,F24: filter @ B,F13: filter @ A,F25: filter @ B,F14: filter @ A] :
( ( filterlim @ A @ B @ F3 @ F24 @ F13 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F24 @ F25 )
=> ( ( ord_less_eq @ ( filter @ A ) @ F14 @ F13 )
=> ( filterlim @ A @ B @ F3 @ F25 @ F14 ) ) ) ) ).
% filterlim_mono
thf(fact_4959_filterlim__inf,axiom,
! [B: $tType,A: $tType,F3: A > B,F24: filter @ B,F33: filter @ B,F13: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( inf_inf @ ( filter @ B ) @ F24 @ F33 ) @ F13 )
= ( ( filterlim @ A @ B @ F3 @ F24 @ F13 )
& ( filterlim @ A @ B @ F3 @ F33 @ F13 ) ) ) ).
% filterlim_inf
thf(fact_4960_tendsto__add__zero,axiom,
! [B: $tType,D: $tType] :
( ( topolo6943815403480290642id_add @ B )
=> ! [F3: D > B,F6: filter @ D,G3: D > B] :
( ( filterlim @ D @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F6 )
=> ( ( filterlim @ D @ B @ G3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F6 )
=> ( filterlim @ D @ B
@ ^ [X4: D] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F6 ) ) ) ) ).
% tendsto_add_zero
thf(fact_4961_tendsto__power,axiom,
! [B: $tType,A: $tType] :
( ( ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F3: A > B,A3: B,F6: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F6 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A3 @ N ) )
@ F6 ) ) ) ).
% tendsto_power
thf(fact_4962_tendsto__power__strong,axiom,
! [B: $tType,C: $tType] :
( ( topolo1898628316856586783d_mult @ B )
=> ! [F3: C > B,A3: B,F6: filter @ C,G3: C > nat,B2: nat] :
( ( filterlim @ C @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F6 )
=> ( ( filterlim @ C @ nat @ G3 @ ( topolo7230453075368039082e_nhds @ nat @ B2 ) @ F6 )
=> ( filterlim @ C @ B
@ ^ [X4: C] : ( power_power @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A3 @ B2 ) )
@ F6 ) ) ) ) ).
% tendsto_power_strong
thf(fact_4963_filterlim__sup,axiom,
! [B: $tType,A: $tType,F3: A > B,F6: filter @ B,F13: filter @ A,F24: filter @ A] :
( ( filterlim @ A @ B @ F3 @ F6 @ F13 )
=> ( ( filterlim @ A @ B @ F3 @ F6 @ F24 )
=> ( filterlim @ A @ B @ F3 @ F6 @ ( sup_sup @ ( filter @ A ) @ F13 @ F24 ) ) ) ) ).
% filterlim_sup
thf(fact_4964_filterlim__ident,axiom,
! [A: $tType,F6: filter @ A] :
( filterlim @ A @ A
@ ^ [X4: A] : X4
@ F6
@ F6 ) ).
% filterlim_ident
thf(fact_4965_filterlim__compose,axiom,
! [B: $tType,A: $tType,C: $tType,G3: A > B,F33: filter @ B,F24: filter @ A,F3: C > A,F13: filter @ C] :
( ( filterlim @ A @ B @ G3 @ F33 @ F24 )
=> ( ( filterlim @ C @ A @ F3 @ F24 @ F13 )
=> ( filterlim @ C @ B
@ ^ [X4: C] : ( G3 @ ( F3 @ X4 ) )
@ F33
@ F13 ) ) ) ).
% filterlim_compose
thf(fact_4966_tendsto__add,axiom,
! [A: $tType,B: $tType] :
( ( topolo6943815403480290642id_add @ A )
=> ! [F3: B > A,A3: A,F6: filter @ B,G3: B > A,B2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F6 )
=> ( ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F6 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A3 @ B2 ) )
@ F6 ) ) ) ) ).
% tendsto_add
thf(fact_4967_tendsto__add__const__iff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [C3: A,F3: B > A,D3: A,F6: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ C3 @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ C3 @ D3 ) )
@ F6 )
= ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ D3 ) @ F6 ) ) ) ).
% tendsto_add_const_iff
thf(fact_4968_filterlim__top,axiom,
! [B: $tType,A: $tType,F3: A > B,F6: filter @ A] : ( filterlim @ A @ B @ F3 @ ( top_top @ ( filter @ B ) ) @ F6 ) ).
% filterlim_top
thf(fact_4969_tendsto__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,A3: B,F6: filter @ A,G3: A > C,B2: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F6 )
=> ( ( filterlim @ A @ C @ G3 @ ( topolo7230453075368039082e_nhds @ C @ B2 ) @ F6 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X4: A] : ( product_Pair @ B @ C @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A3 @ B2 ) )
@ F6 ) ) ) ) ).
% tendsto_Pair
thf(fact_4970_tendsto__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: A > B,L: filter @ B,X3: A,S3: set @ A,T4: set @ A] :
( ( filterlim @ A @ B @ F3 @ L @ ( topolo174197925503356063within @ A @ X3 @ S3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T4 @ S3 )
=> ( filterlim @ A @ B @ F3 @ L @ ( topolo174197925503356063within @ A @ X3 @ T4 ) ) ) ) ) ).
% tendsto_within_subset
thf(fact_4971_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_4972_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_4973_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_4974_tendsto__null__power,axiom,
! [B: $tType,A: $tType] :
( ( real_V2822296259951069270ebra_1 @ B )
=> ! [F3: A > B,F6: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F6 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F6 ) ) ) ) ).
% tendsto_null_power
thf(fact_4975_LIM__offset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,L5: B,A3: A,K2: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( F3 @ ( plus_plus @ A @ X4 @ K2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L5 )
@ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A3 @ K2 ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset
thf(fact_4976_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U )
= ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_4977_extract__Nil__code,axiom,
! [A: $tType,P: A > $o] :
( ( extract @ A @ P @ ( nil @ A ) )
= ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
% extract_Nil_code
thf(fact_4978_extract__None__iff,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( ( extract @ A @ P @ Xs2 )
= ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) )
= ( ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
& ( P @ X4 ) ) ) ) ).
% extract_None_iff
thf(fact_4979_filterlim__INF_H,axiom,
! [C: $tType,B: $tType,A: $tType,X3: A,A6: set @ A,F3: B > C,F6: filter @ C,G7: A > ( filter @ B )] :
( ( member @ A @ X3 @ A6 )
=> ( ( filterlim @ B @ C @ F3 @ F6 @ ( G7 @ X3 ) )
=> ( filterlim @ B @ C @ F3 @ F6 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ G7 @ A6 ) ) ) ) ) ).
% filterlim_INF'
thf(fact_4980_filterlim__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,G7: C > ( filter @ B ),B5: set @ C,F6: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ G7 @ B5 ) ) @ F6 )
= ( ! [X4: C] :
( ( member @ C @ X4 @ B5 )
=> ( filterlim @ A @ B @ F3 @ ( G7 @ X4 ) @ F6 ) ) ) ) ).
% filterlim_INF
thf(fact_4981_DERIV__LIM__iff,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > A,A3: A,D4: A] :
( ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ A3 @ H2 ) ) @ ( F3 @ A3 ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [X4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ X4 ) @ ( F3 @ A3 ) ) @ ( minus_minus @ A @ X4 @ A3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_LIM_iff
thf(fact_4982_DERIV__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X3: A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X3 @ H2 ) ) @ ( F3 @ X3 ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_def
thf(fact_4983_DERIV__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X3: A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X3 @ H2 ) ) @ ( F3 @ X3 ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_D
thf(fact_4984_field__has__derivative__at,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X3: A] :
( ( has_derivative @ A @ A @ F3 @ ( times_times @ A @ D4 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X3 @ H2 ) ) @ ( F3 @ X3 ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% field_has_derivative_at
thf(fact_4985_filterlim__at__to__0,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: A > B,F6: filter @ B,A3: A] :
( ( filterlim @ A @ B @ F3 @ F6 @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [X4: A] : ( F3 @ ( plus_plus @ A @ X4 @ A3 ) )
@ F6
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_at_to_0
thf(fact_4986_filterlim__shift__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: A > B,D3: A,F6: filter @ B,A3: A] :
( ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F3 @ ( plus_plus @ A @ D3 ) ) @ F6 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A3 @ D3 ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B @ F3 @ F6 @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_shift_iff
thf(fact_4987_filterlim__shift,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: A > B,F6: filter @ B,A3: A,D3: A] :
( ( filterlim @ A @ B @ F3 @ F6 @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F3 @ ( plus_plus @ A @ D3 ) ) @ F6 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A3 @ D3 ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_shift
thf(fact_4988_powser__limit__0__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A3: nat > A,F3: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X5: A] :
( ( X5
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ S )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A3 @ N3 ) @ ( power_power @ A @ X5 @ N3 ) )
@ ( F3 @ X5 ) ) ) )
=> ( 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_4989_powser__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A3: nat > A,F3: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X5: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ S )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A3 @ N3 ) @ ( power_power @ A @ X5 @ N3 ) )
@ ( F3 @ X5 ) ) )
=> ( 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_4990_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_4991_LIM__cos__div__sin,axiom,
( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( cos @ real @ X4 ) @ ( sin @ real @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( top_top @ ( set @ real ) ) ) ) ).
% LIM_cos_div_sin
thf(fact_4992_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_4993_extract__Cons__code,axiom,
! [A: $tType,P: A > $o,X3: A,Xs2: list @ A] :
( ( ( P @ X3 )
=> ( ( extract @ A @ P @ ( cons @ A @ X3 @ Xs2 ) )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( nil @ A ) @ ( product_Pair @ A @ ( list @ A ) @ X3 @ Xs2 ) ) ) ) )
& ( ~ ( P @ X3 )
=> ( ( extract @ A @ P @ ( cons @ A @ X3 @ Xs2 ) )
= ( case_option @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Ys3: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y3: A,Zs3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( cons @ A @ X3 @ Ys3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y3 @ Zs3 ) ) ) ) )
@ ( extract @ A @ P @ Xs2 ) ) ) ) ) ).
% extract_Cons_code
thf(fact_4994_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 ) )
=> ! [N9: 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 ) ) @ N9 ) @ ( 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 ) ) @ N9 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_4995_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 ) ) )
=> ! [N9: 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 ) ) @ N9 ) ) )
@ ( 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 ) ) @ N9 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_4996_summable__Leibniz_H_I4_J,axiom,
! [A3: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( ord_less_eq @ real
@ ( suminf @ real
@ ^ [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_4997_trivial__limit__sequentially,axiom,
( ( at_top @ nat )
!= ( bot_bot @ ( filter @ nat ) ) ) ).
% trivial_limit_sequentially
thf(fact_4998_filterlim__Suc,axiom,
filterlim @ nat @ nat @ suc @ ( at_top @ nat ) @ ( at_top @ nat ) ).
% filterlim_Suc
thf(fact_4999_filterlim__sequentially__Suc,axiom,
! [A: $tType,F3: nat > A,F6: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X4: nat] : ( F3 @ ( suc @ X4 ) )
@ F6
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F3 @ F6 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_5000_trivial__limit__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_top @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_top_linorder
thf(fact_5001_LIMSEQ__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,L: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F3 @ ( suc @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_Suc
thf(fact_5002_LIMSEQ__imp__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,L: A] :
( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F3 @ ( suc @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_imp_Suc
thf(fact_5003_LIMSEQ__ignore__initial__segment,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,A3: A,K2: nat] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F3 @ ( plus_plus @ nat @ N3 @ K2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_5004_LIMSEQ__offset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,K2: nat,A3: A] :
( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F3 @ ( plus_plus @ nat @ N3 @ K2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_offset
thf(fact_5005_LIMSEQ__le__const2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X6: nat > A,X3: A,A3: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ? [N7: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N7 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ N2 ) @ A3 ) )
=> ( ord_less_eq @ A @ X3 @ A3 ) ) ) ) ).
% LIMSEQ_le_const2
thf(fact_5006_LIMSEQ__le__const,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X6: nat > A,X3: A,A3: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ? [N7: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N7 @ N2 )
=> ( ord_less_eq @ A @ A3 @ ( X6 @ N2 ) ) )
=> ( ord_less_eq @ A @ A3 @ X3 ) ) ) ) ).
% LIMSEQ_le_const
thf(fact_5007_Lim__bounded2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: nat > A,L: A,N5: nat,C4: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ord_less_eq @ A @ C4 @ ( F3 @ N2 ) ) )
=> ( ord_less_eq @ A @ C4 @ L ) ) ) ) ).
% Lim_bounded2
thf(fact_5008_Lim__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: nat > A,L: A,M7: nat,C4: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ M7 @ N2 )
=> ( ord_less_eq @ A @ ( F3 @ N2 ) @ C4 ) )
=> ( ord_less_eq @ A @ L @ C4 ) ) ) ) ).
% Lim_bounded
thf(fact_5009_LIMSEQ__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X6: nat > A,X3: A,Y8: nat > A,Y: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y8 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ? [N7: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N7 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ) ).
% LIMSEQ_le
thf(fact_5010_lim__mono,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [N5: nat,X6: nat > A,Y8: nat > A,X3: A,Y: A] :
( ! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) )
=> ( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y8 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ) ).
% lim_mono
thf(fact_5011_Sup__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: nat > A,S: set @ A,A3: A] :
( ! [N2: nat] : ( member @ A @ ( B2 @ N2 ) @ S )
=> ( ( filterlim @ nat @ A @ B2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ A3 @ ( complete_Sup_Sup @ A @ S ) ) ) ) ) ).
% Sup_lim
thf(fact_5012_Inf__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: nat > A,S: set @ A,A3: A] :
( ! [N2: nat] : ( member @ A @ ( B2 @ N2 ) @ S )
=> ( ( filterlim @ nat @ A @ B2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ S ) @ A3 ) ) ) ) ).
% Inf_lim
thf(fact_5013_Inf__as__limit,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A6: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ? [U4: nat > A] :
( ! [N9: nat] : ( member @ A @ ( U4 @ N9 ) @ A6 )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ ( complete_Inf_Inf @ A @ A6 ) ) @ ( at_top @ nat ) ) ) ) ) ).
% Inf_as_limit
thf(fact_5014_monoseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: nat > A,X3: A] :
( ( topological_monoseq @ A @ A3 )
=> ( ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ( ! [N9: nat] : ( ord_less_eq @ A @ ( A3 @ N9 ) @ X3 )
& ! [M3: nat,N9: nat] :
( ( ord_less_eq @ nat @ M3 @ N9 )
=> ( ord_less_eq @ A @ ( A3 @ M3 ) @ ( A3 @ N9 ) ) ) )
| ( ! [N9: nat] : ( ord_less_eq @ A @ X3 @ ( A3 @ N9 ) )
& ! [M3: nat,N9: nat] :
( ( ord_less_eq @ nat @ M3 @ N9 )
=> ( ord_less_eq @ A @ ( A3 @ N9 ) @ ( A3 @ M3 ) ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_5015_telescope__summable,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C3: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N3 ) ) @ ( F3 @ N3 ) ) ) ) ) ).
% telescope_summable
thf(fact_5016_telescope__summable_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C3: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) ) ) ) ) ).
% telescope_summable'
thf(fact_5017_nested__sequence__unique,axiom,
! [F3: nat > real,G3: nat > real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( G3 @ ( suc @ N2 ) ) @ ( G3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( F3 @ N2 ) @ ( G3 @ N2 ) )
=> ( ( filterlim @ nat @ real
@ ^ [N3: nat] : ( minus_minus @ real @ ( F3 @ N3 ) @ ( G3 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) )
=> ? [L4: real] :
( ! [N9: nat] : ( ord_less_eq @ real @ ( F3 @ N9 ) @ L4 )
& ( filterlim @ nat @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L4 ) @ ( at_top @ nat ) )
& ! [N9: nat] : ( ord_less_eq @ real @ L4 @ ( G3 @ N9 ) )
& ( filterlim @ nat @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ L4 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_5018_LIMSEQ__inverse__zero,axiom,
! [X6: nat > real] :
( ! [R3: real] :
? [N7: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N7 @ N2 )
=> ( ord_less @ real @ R3 @ ( X6 @ N2 ) ) )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( inverse_inverse @ real @ ( X6 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_zero
thf(fact_5019_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim @ nat @ real
@ ^ [N3: nat] : ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_5020_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( plus_plus @ real @ R2 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_5021_increasing__LIMSEQ,axiom,
! [F3: nat > real,L: real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( F3 @ N2 ) @ L )
=> ( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [N9: nat] : ( ord_less_eq @ real @ L @ ( plus_plus @ real @ ( F3 @ N9 ) @ E2 ) ) )
=> ( filterlim @ nat @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( at_top @ nat ) ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_5022_LIMSEQ__n__over__Suc__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_n_over_Suc_n
thf(fact_5023_LIMSEQ__Suc__n__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) @ ( semiring_1_of_nat @ A @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_Suc_n_over_n
thf(fact_5024_LIMSEQ__realpow__zero,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ X3 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_realpow_zero
thf(fact_5025_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
@ ^ [N3: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N3 ) ) @ ( F3 @ N3 ) )
@ ( minus_minus @ A @ C3 @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% telescope_sums
thf(fact_5026_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
@ ^ [N3: nat] : ( minus_minus @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
@ ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ C3 ) ) ) ) ).
% telescope_sums'
thf(fact_5027_LIMSEQ__divide__realpow__zero,axiom,
! [X3: real,A3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( divide_divide @ real @ A3 @ ( power_power @ real @ X3 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_5028_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_5029_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_5030_LIMSEQ__inverse__realpow__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( inverse_inverse @ real @ ( power_power @ real @ X3 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_5031_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( plus_plus @ real @ R2 @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_5032_LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A,L5: A,R2: real] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ No @ N9 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ N9 ) @ L5 ) ) @ R2 ) ) ) ) ) ).
% LIMSEQ_D
thf(fact_5033_LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A,L5: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No2: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No2 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ N2 ) @ L5 ) ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_I
thf(fact_5034_LIMSEQ__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No3 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ N3 ) @ L5 ) ) @ R5 ) ) ) ) ) ) ).
% LIMSEQ_iff
thf(fact_5035_LIMSEQ__power__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X3 ) @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_power_zero
thf(fact_5036_tendsto__exp__limit__sequentially,axiom,
! [X3: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X3 @ ( semiring_1_of_nat @ real @ N3 ) ) ) @ N3 )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X3 ) )
@ ( at_top @ nat ) ) ).
% tendsto_exp_limit_sequentially
thf(fact_5037_tendsto__at__iff__sequentially,axiom,
! [C: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > C,A3: C,X3: A,S: set @ A] :
( ( filterlim @ A @ C @ F3 @ ( topolo7230453075368039082e_nhds @ C @ A3 ) @ ( topolo174197925503356063within @ A @ X3 @ S ) )
= ( ! [X8: nat > A] :
( ! [I4: nat] : ( member @ A @ ( X8 @ I4 ) @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( 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_5038_tendsto__power__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F3: B > nat,F6: filter @ B,X3: A] :
( ( filterlim @ B @ nat @ F3 @ ( at_top @ nat ) @ F6 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) )
=> ( filterlim @ B @ A
@ ^ [Y3: B] : ( power_power @ A @ X3 @ ( F3 @ Y3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F6 ) ) ) ) ).
% tendsto_power_zero
thf(fact_5039_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( times_times @ real @ R2 @ ( plus_plus @ real @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_5040_LIMSEQ__norm__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ! [N2: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_norm_0
thf(fact_5041_summable__Leibniz_I1_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( summable @ real
@ ^ [N3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( A3 @ N3 ) ) ) ) ) ).
% summable_Leibniz(1)
thf(fact_5042_field__derivative__lim__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Df: A,Z2: A,S: nat > A,A3: A] :
( ( has_field_derivative @ A @ F3 @ Df @ ( topolo174197925503356063within @ A @ Z2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ nat @ A @ S @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( S @ N2 )
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ Z2 @ ( S @ N3 ) ) ) @ ( F3 @ Z2 ) ) @ ( S @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( at_top @ nat ) )
=> ( Df = A3 ) ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_5043_powser__times__n__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ X3 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% powser_times_n_limit_0
thf(fact_5044_lim__n__over__pown,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ X3 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% lim_n_over_pown
thf(fact_5045_summable,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( summable @ real
@ ^ [N3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( A3 @ N3 ) ) ) ) ) ) ).
% summable
thf(fact_5046_cos__diff__limit__1,axiom,
! [Theta: nat > real,Theta2: real] :
( ( filterlim @ nat @ real
@ ^ [J3: nat] : ( cos @ real @ ( minus_minus @ real @ ( Theta @ J3 ) @ Theta2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ~ ! [K: nat > int] :
~ ( filterlim @ nat @ real
@ ^ [J3: nat] : ( minus_minus @ real @ ( Theta @ J3 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K @ J3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ Theta2 )
@ ( at_top @ nat ) ) ) ).
% cos_diff_limit_1
thf(fact_5047_cos__limit__1,axiom,
! [Theta: nat > real] :
( ( filterlim @ nat @ real
@ ^ [J3: nat] : ( cos @ real @ ( Theta @ J3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ? [K: nat > int] :
( filterlim @ nat @ real
@ ^ [J3: nat] : ( minus_minus @ real @ ( Theta @ J3 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K @ J3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% cos_limit_1
thf(fact_5048_summable__Leibniz_I4_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( filterlim @ nat @ real
@ ^ [N3: 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 ) ) @ N3 ) ) )
@ ( 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(4)
thf(fact_5049_zeroseq__arctan__series,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% zeroseq_arctan_series
thf(fact_5050_summable__Leibniz_H_I3_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( filterlim @ nat @ real
@ ^ [N3: 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 ) ) @ N3 ) ) )
@ ( 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_5051_summable__Leibniz_H_I2_J,axiom,
! [A3: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [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_5052_sums__alternating__upper__lower,axiom,
! [A3: nat > real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ? [L4: real] :
( ! [N9: 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 ) ) @ N9 ) ) )
@ L4 )
& ( filterlim @ nat @ real
@ ^ [N3: 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 ) ) @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L4 )
@ ( at_top @ nat ) )
& ! [N9: nat] :
( ord_less_eq @ real @ L4
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) @ ( one_one @ nat ) ) ) ) )
& ( filterlim @ nat @ real
@ ^ [N3: 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 ) ) @ N3 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L4 )
@ ( at_top @ nat ) ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_5053_summable__Leibniz_I5_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( filterlim @ nat @ real
@ ^ [N3: 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 ) ) @ N3 ) @ ( 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_5054_summable__Leibniz_H_I5_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( filterlim @ nat @ real
@ ^ [N3: 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 ) ) @ N3 ) @ ( 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_5055_has__derivative__at2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F8: A > B,X3: A] :
( ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ( filterlim @ A @ B
@ ^ [Y3: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ X3 ) ) ) @ ( minus_minus @ B @ ( F3 @ Y3 ) @ ( plus_plus @ B @ ( F3 @ X3 ) @ ( F8 @ ( minus_minus @ A @ Y3 @ X3 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at2
thf(fact_5056_has__derivative__at,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,D4: A > B,X3: A] :
( ( has_derivative @ A @ B @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ D4 )
& ( filterlim @ A @ real
@ ^ [H2: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ ( plus_plus @ A @ X3 @ H2 ) ) @ ( F3 @ X3 ) ) @ ( D4 @ 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_5057_has__derivative__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F8: A > B,X3: A,S: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ( filterlim @ A @ B
@ ^ [Y3: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ X3 ) ) ) @ ( minus_minus @ B @ ( F3 @ Y3 ) @ ( plus_plus @ B @ ( F3 @ X3 ) @ ( F8 @ ( minus_minus @ A @ Y3 @ X3 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% has_derivative_within
thf(fact_5058_bounded__linear__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( real_V3181309239436604168linear @ A @ B @ G3 )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X4: A] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ).
% bounded_linear_add
thf(fact_5059_has__derivative__within__singleton__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G3: A > B,X3: A] :
( ( has_derivative @ A @ B @ F3 @ G3 @ ( topolo174197925503356063within @ A @ X3 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( real_V3181309239436604168linear @ A @ B @ G3 ) ) ) ).
% has_derivative_within_singleton_iff
thf(fact_5060_filterlim__pow__at__top,axiom,
! [A: $tType,N: nat,F3: A > real,F6: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F6 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( power_power @ real @ ( F3 @ X4 ) @ N )
@ ( at_top @ real )
@ F6 ) ) ) ).
% filterlim_pow_at_top
thf(fact_5061_bounded__linear__intro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,K5: real] :
( ! [X5: A,Y4: A] :
( ( F3 @ ( plus_plus @ A @ X5 @ Y4 ) )
= ( plus_plus @ B @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) )
=> ( ! [R3: real,X5: A] :
( ( F3 @ ( real_V8093663219630862766scaleR @ A @ R3 @ X5 ) )
= ( real_V8093663219630862766scaleR @ B @ R3 @ ( F3 @ X5 ) ) )
=> ( ! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K5 ) )
=> ( real_V3181309239436604168linear @ A @ B @ F3 ) ) ) ) ) ).
% bounded_linear_intro
thf(fact_5062_tendsto__power__div__exp__0,axiom,
! [K2: nat] :
( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( power_power @ real @ X4 @ K2 ) @ ( exp @ real @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% tendsto_power_div_exp_0
thf(fact_5063_filterlim__tan__at__left,axiom,
filterlim @ real @ real @ ( tan @ real ) @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( set_ord_lessThan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% filterlim_tan_at_left
thf(fact_5064_tendsto__arctan__at__top,axiom,
filterlim @ real @ real @ arctan @ ( topolo7230453075368039082e_nhds @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( at_top @ real ) ).
% tendsto_arctan_at_top
thf(fact_5065_has__derivative__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F8: A > B,X3: A] :
( ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ? [E4: A > B] :
( ! [H2: A] :
( ( F3 @ ( plus_plus @ A @ X3 @ H2 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F3 @ X3 ) @ ( F8 @ H2 ) ) @ ( E4 @ H2 ) ) )
& ( filterlim @ A @ real
@ ^ [H2: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E4 @ 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_5066_has__derivative__at__within__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [X3: A,S3: set @ A,F3: A > B,F8: A > B] :
( ( member @ A @ X3 @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X3 @ S3 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ? [E4: A > B] :
( ! [H2: A] :
( ( member @ A @ ( plus_plus @ A @ X3 @ H2 ) @ S3 )
=> ( ( F3 @ ( plus_plus @ A @ X3 @ H2 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F3 @ X3 ) @ ( F8 @ H2 ) ) @ ( E4 @ H2 ) ) ) )
& ( filterlim @ A @ real
@ ^ [H2: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E4 @ 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_5067_tendsto__arctan__at__bot,axiom,
filterlim @ real @ real @ arctan @ ( topolo7230453075368039082e_nhds @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) @ ( at_bot @ real ) ).
% tendsto_arctan_at_bot
thf(fact_5068_filterlim__pow__at__bot__even,axiom,
! [N: nat,F3: real > real,F6: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F3 @ ( at_bot @ real ) @ F6 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( power_power @ real @ ( F3 @ X4 ) @ N )
@ ( at_top @ real )
@ F6 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_5069_open__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo1002775350975398744n_open @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% open_empty
thf(fact_5070_openI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ? [T8: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T8 )
& ( member @ A @ X5 @ T8 )
& ( ord_less_eq @ ( set @ A ) @ T8 @ S3 ) ) )
=> ( topolo1002775350975398744n_open @ A @ S3 ) ) ) ).
% openI
thf(fact_5071_open__subopen,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ S6 )
=> ? [T9: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T9 )
& ( member @ A @ X4 @ T9 )
& ( ord_less_eq @ ( set @ A ) @ T9 @ S6 ) ) ) ) ) ) ).
% open_subopen
thf(fact_5072_first__countable__basis,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X3: A] :
? [A10: nat > ( set @ A )] :
( ! [I2: nat] :
( ( member @ A @ X3 @ ( A10 @ I2 ) )
& ( topolo1002775350975398744n_open @ A @ ( A10 @ I2 ) ) )
& ! [S9: set @ A] :
( ( ( topolo1002775350975398744n_open @ A @ S9 )
& ( member @ A @ X3 @ S9 ) )
=> ? [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( A10 @ I3 ) @ S9 ) ) ) ) ).
% first_countable_basis
thf(fact_5073_not__open__singleton,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [X3: A] :
~ ( topolo1002775350975398744n_open @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% not_open_singleton
thf(fact_5074_hausdorff,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
=> ? [U5: set @ A,V6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U5 )
& ( topolo1002775350975398744n_open @ A @ V6 )
& ( member @ A @ X3 @ U5 )
& ( member @ A @ Y @ V6 )
& ( ( inf_inf @ ( set @ A ) @ U5 @ V6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% hausdorff
thf(fact_5075_separation__t2,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
= ( ? [U6: set @ A,V7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U6 )
& ( topolo1002775350975398744n_open @ A @ V7 )
& ( member @ A @ X3 @ U6 )
& ( member @ A @ Y @ V7 )
& ( ( inf_inf @ ( set @ A ) @ U6 @ V7 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% separation_t2
thf(fact_5076_trivial__limit__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_bot @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_bot_linorder
thf(fact_5077_at__within__open__subset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: A,S3: set @ A,T4: set @ A] :
( ( member @ A @ A3 @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ T4 )
=> ( ( topolo174197925503356063within @ A @ A3 @ T4 )
= ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% at_within_open_subset
thf(fact_5078_open__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A,X3: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( member @ A @ X3 @ S3 )
=> ( ( ord_less @ A @ X3 @ Y )
=> ? [B4: A] :
( ( ord_less @ A @ X3 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ X3 @ B4 ) @ S3 ) ) ) ) ) ) ).
% open_right
thf(fact_5079_lim__explicit,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,F0: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ F0 ) @ ( at_top @ nat ) )
= ( ! [S6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S6 )
=> ( ( member @ A @ F0 @ S6 )
=> ? [N6: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( member @ A @ ( F3 @ N3 ) @ S6 ) ) ) ) ) ) ) ).
% lim_explicit
thf(fact_5080_at__within__nhd,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X3: A,S3: set @ A,T4: set @ A,U3: set @ A] :
( ( member @ A @ X3 @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ T4 @ S3 ) @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ U3 @ S3 ) @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( topolo174197925503356063within @ A @ X3 @ T4 )
= ( topolo174197925503356063within @ A @ X3 @ U3 ) ) ) ) ) ) ).
% at_within_nhd
thf(fact_5081_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_5082_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F3: real > real,F6: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F3 @ ( at_bot @ real ) @ F6 )
=> ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( power_power @ real @ ( F3 @ X4 ) @ N )
@ ( at_bot @ real )
@ F6 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_5083_tendsto__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A3: A,S3: set @ A,F3: A > D,L5: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A3 )
=> ( ( member @ A @ A3 @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( filterlim @ A @ D @ F3 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ S3 ) )
= ( 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_5084_filterlim__tan__at__right,axiom,
filterlim @ real @ real @ ( tan @ real ) @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% filterlim_tan_at_right
thf(fact_5085_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_5086_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X3 ) @ ( set_ord_greaterThan @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% greaterThan_subset_iff
thf(fact_5087_greaterThan__non__empty,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X3: A] :
( ( set_ord_greaterThan @ A @ X3 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% greaterThan_non_empty
thf(fact_5088_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_5089_less__separate,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ? [A5: A,B4: A] :
( ( member @ A @ X3 @ ( set_ord_lessThan @ A @ A5 ) )
& ( member @ A @ Y @ ( set_ord_greaterThan @ A @ B4 ) )
& ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A5 ) @ ( set_ord_greaterThan @ A @ B4 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% less_separate
thf(fact_5090_scale__right__distrib__NO__MATCH,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A,Y: A,A3: real] :
( ( nO_MATCH @ A @ real @ ( divide_divide @ A @ X3 @ Y ) @ A3 )
=> ( ( real_V8093663219630862766scaleR @ A @ A3 @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y ) ) ) ) ) ).
% scale_right_distrib_NO_MATCH
thf(fact_5091_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X3: B,Y: B,A3: A,B2: A,C3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_5092_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X3: B,Y: B,C3: A,A3: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ C3 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_5093_power__minus_H,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: A,N: nat] :
( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X3 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X3 @ N ) ) ) ) ) ).
% power_minus'
thf(fact_5094_scale__left__distrib__NO__MATCH,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A,Y: A,C3: C,A3: real,B2: real] :
( ( nO_MATCH @ A @ C @ ( divide_divide @ A @ X3 @ Y ) @ C3 )
=> ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A3 @ B2 ) @ X3 )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) ) ) ) ).
% scale_left_distrib_NO_MATCH
thf(fact_5095_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X3: B,B2: A,A3: A] :
( ( nO_MATCH @ B @ A @ X3 @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_5096_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X3: B,B2: A,A3: A] :
( ( nO_MATCH @ B @ A @ X3 @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_5097_at__within__order,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X3: A,S: set @ A] :
( ( ( top_top @ ( set @ A ) )
!= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( topolo174197925503356063within @ A @ X3 @ S )
= ( inf_inf @ ( filter @ A )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [A8: A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A8 ) @ S ) @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( set_ord_greaterThan @ A @ X3 ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [A8: A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A8 ) @ S ) @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( set_ord_lessThan @ A @ X3 ) ) ) ) ) ) ) ).
% at_within_order
thf(fact_5098_principal__inject,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ( principal @ A @ A6 )
= ( principal @ A @ B5 ) )
= ( A6 = B5 ) ) ).
% principal_inject
thf(fact_5099_principal__le__iff,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ ( principal @ A @ A6 ) @ ( principal @ A @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A6 @ B5 ) ) ).
% principal_le_iff
thf(fact_5100_inf__principal,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( inf_inf @ ( filter @ A ) @ ( principal @ A @ A6 ) @ ( principal @ A @ B5 ) )
= ( principal @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ) ).
% inf_principal
thf(fact_5101_sup__principal,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( sup_sup @ ( filter @ A ) @ ( principal @ A @ A6 ) @ ( principal @ A @ B5 ) )
= ( principal @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) ) ) ).
% sup_principal
thf(fact_5102_SUP__principal,axiom,
! [A: $tType,B: $tType,A6: B > ( set @ A ),I5: set @ B] :
( ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [I4: B] : ( principal @ A @ ( A6 @ I4 ) )
@ I5 ) )
= ( principal @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I5 ) ) ) ) ).
% SUP_principal
thf(fact_5103_principal__eq__bot__iff,axiom,
! [A: $tType,X6: set @ A] :
( ( ( principal @ A @ X6 )
= ( bot_bot @ ( filter @ A ) ) )
= ( X6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% principal_eq_bot_iff
thf(fact_5104_bot__eq__principal__empty,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( principal @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bot_eq_principal_empty
thf(fact_5105_top__eq__principal__UNIV,axiom,
! [A: $tType] :
( ( top_top @ ( filter @ A ) )
= ( principal @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% top_eq_principal_UNIV
thf(fact_5106_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_5107_filterlim__If,axiom,
! [B: $tType,A: $tType,F3: A > B,G7: filter @ B,F6: filter @ A,P: A > $o,G3: A > B] :
( ( filterlim @ A @ B @ F3 @ G7 @ ( inf_inf @ ( filter @ A ) @ F6 @ ( principal @ A @ ( collect @ A @ P ) ) ) )
=> ( ( filterlim @ A @ B @ G3 @ G7
@ ( inf_inf @ ( filter @ A ) @ F6
@ ( principal @ A
@ ( collect @ A
@ ^ [X4: A] :
~ ( P @ X4 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( if @ B @ ( P @ X4 ) @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ G7
@ F6 ) ) ) ).
% filterlim_If
thf(fact_5108_nhds__discrete,axiom,
! [A: $tType] :
( ( topolo8865339358273720382pology @ A )
=> ( ( topolo7230453075368039082e_nhds @ A )
= ( ^ [X4: A] : ( principal @ A @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% nhds_discrete
thf(fact_5109_filterlim__base,axiom,
! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,J5: set @ A,I: A > C,I5: set @ C,F6: C > ( set @ D ),F3: D > E,G7: A > ( set @ E )] :
( ! [M: A,X5: B] :
( ( member @ A @ M @ J5 )
=> ( member @ C @ ( I @ M ) @ I5 ) )
=> ( ! [M: A,X5: D] :
( ( member @ A @ M @ J5 )
=> ( ( member @ D @ X5 @ ( F6 @ ( I @ M ) ) )
=> ( member @ E @ ( F3 @ X5 ) @ ( G7 @ M ) ) ) )
=> ( filterlim @ D @ E @ F3
@ ( complete_Inf_Inf @ ( filter @ E )
@ ( image2 @ A @ ( filter @ E )
@ ^ [J3: A] : ( principal @ E @ ( G7 @ J3 ) )
@ J5 ) )
@ ( complete_Inf_Inf @ ( filter @ D )
@ ( image2 @ C @ ( filter @ D )
@ ^ [I4: C] : ( principal @ D @ ( F6 @ I4 ) )
@ I5 ) ) ) ) ) ).
% filterlim_base
thf(fact_5110_tendsto__principal__singleton,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,X3: B] : ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( F3 @ X3 ) ) @ ( principal @ B @ ( insert2 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% tendsto_principal_singleton
thf(fact_5111_nhds__discrete__open,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X3: A] :
( ( topolo1002775350975398744n_open @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( topolo7230453075368039082e_nhds @ A @ X3 )
= ( principal @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% nhds_discrete_open
thf(fact_5112_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image2 @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_5113_greaterThan__Suc,axiom,
! [K2: nat] :
( ( set_ord_greaterThan @ nat @ ( suc @ K2 ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K2 ) @ ( insert2 @ nat @ ( suc @ K2 ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% greaterThan_Suc
thf(fact_5114_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_5115_filterlim__base__iff,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,I5: set @ A,F6: A > ( set @ B ),F3: B > C,G7: D > ( set @ C ),J5: set @ D] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( F6 @ I3 ) @ ( F6 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( F6 @ J2 ) @ ( F6 @ I3 ) ) ) ) )
=> ( ( filterlim @ B @ C @ F3
@ ( complete_Inf_Inf @ ( filter @ C )
@ ( image2 @ D @ ( filter @ C )
@ ^ [J3: D] : ( principal @ C @ ( G7 @ J3 ) )
@ J5 ) )
@ ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [I4: A] : ( principal @ B @ ( F6 @ I4 ) )
@ I5 ) ) )
= ( ! [X4: D] :
( ( member @ D @ X4 @ J5 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ I5 )
& ! [Z4: B] :
( ( member @ B @ Z4 @ ( F6 @ Y3 ) )
=> ( member @ C @ ( F3 @ Z4 ) @ ( G7 @ X4 ) ) ) ) ) ) ) ) ) ).
% filterlim_base_iff
thf(fact_5116_INF__principal__finite,axiom,
! [B: $tType,A: $tType,X6: set @ A,F3: A > ( set @ B )] :
( ( finite_finite2 @ A @ X6 )
=> ( ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [X4: A] : ( principal @ B @ ( F3 @ X4 ) )
@ X6 ) )
= ( principal @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ X6 ) ) ) ) ) ).
% INF_principal_finite
thf(fact_5117_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_5118_at__within__def,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [A8: A,S7: set @ A] : ( inf_inf @ ( filter @ A ) @ ( topolo7230453075368039082e_nhds @ A @ A8 ) @ ( principal @ A @ ( minus_minus @ ( set @ A ) @ S7 @ ( insert2 @ A @ A8 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% at_within_def
thf(fact_5119_at__within__eq,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [X4: A,S7: set @ A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ ( set @ A ) @ ( filter @ A )
@ ^ [S6: set @ A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S6 @ S7 ) @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [S6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S6 )
& ( member @ A @ X4 @ S6 ) ) ) ) ) ) ) ) ).
% at_within_eq
thf(fact_5120_rat__inverse__code,axiom,
! [P2: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A8: int,B8: int] :
( if @ ( product_prod @ int @ int )
@ ( A8
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A8 ) @ B8 ) @ ( abs_abs @ int @ A8 ) ) )
@ ( quotient_of @ P2 ) ) ) ).
% rat_inverse_code
thf(fact_5121_normalize__negative,axiom,
! [Q3: int,P2: int] :
( ( ord_less @ int @ Q3 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P2 @ Q3 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P2 ) @ ( uminus_uminus @ int @ Q3 ) ) ) ) ) ).
% normalize_negative
thf(fact_5122_filterlim__realpow__sequentially__gt1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X3: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X3 ) @ ( at_infinity @ A ) @ ( at_top @ nat ) ) ) ) ).
% filterlim_realpow_sequentially_gt1
thf(fact_5123_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_5124_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_5125_quotient__of__number_I3_J,axiom,
! [K2: num] :
( ( quotient_of @ ( numeral_numeral @ rat @ K2 ) )
= ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K2 ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(3)
thf(fact_5126_quotient__of__number_I4_J,axiom,
( ( quotient_of @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
= ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(4)
thf(fact_5127_normalize__denom__zero,axiom,
! [P2: int] :
( ( normalize @ ( product_Pair @ int @ int @ P2 @ ( zero_zero @ int ) ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% normalize_denom_zero
thf(fact_5128_quotient__of__number_I5_J,axiom,
! [K2: num] :
( ( quotient_of @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K2 ) ) )
= ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(5)
thf(fact_5129_quotient__of__div,axiom,
! [R2: rat,N: int,D3: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair @ int @ int @ N @ D3 ) )
=> ( R2
= ( divide_divide @ rat @ ( ring_1_of_int @ rat @ N ) @ ( ring_1_of_int @ rat @ D3 ) ) ) ) ).
% quotient_of_div
thf(fact_5130_rat__divide__code,axiom,
! [P2: rat,Q3: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P2 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A8: int,C6: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B8: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A8 @ D5 ) @ ( times_times @ int @ C6 @ B8 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P2 ) ) ) ).
% rat_divide_code
thf(fact_5131_rat__times__code,axiom,
! [P2: rat,Q3: rat] :
( ( quotient_of @ ( times_times @ rat @ P2 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A8: int,C6: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B8: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A8 @ B8 ) @ ( times_times @ int @ C6 @ D5 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P2 ) ) ) ).
% rat_times_code
thf(fact_5132_rat__plus__code,axiom,
! [P2: rat,Q3: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P2 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A8: int,C6: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B8: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A8 @ D5 ) @ ( times_times @ int @ B8 @ C6 ) ) @ ( times_times @ int @ C6 @ D5 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P2 ) ) ) ).
% rat_plus_code
thf(fact_5133_rat__minus__code,axiom,
! [P2: rat,Q3: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P2 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A8: int,C6: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B8: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A8 @ D5 ) @ ( times_times @ int @ B8 @ C6 ) ) @ ( times_times @ int @ C6 @ D5 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P2 ) ) ) ).
% rat_minus_code
thf(fact_5134_quotient__of__denom__pos,axiom,
! [R2: rat,P2: int,Q3: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair @ int @ int @ P2 @ Q3 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% quotient_of_denom_pos
thf(fact_5135_tendsto__add__filterlim__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,C3: B,F6: filter @ A,G3: A > B] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ C3 ) @ F6 )
=> ( ( filterlim @ A @ B @ G3 @ ( at_infinity @ B ) @ F6 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( at_infinity @ B )
@ F6 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity
thf(fact_5136_tendsto__add__filterlim__at__infinity_H,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F6: filter @ A,G3: A > B,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F6 )
=> ( ( filterlim @ A @ B @ G3 @ ( topolo7230453075368039082e_nhds @ B @ C3 ) @ F6 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( at_infinity @ B )
@ F6 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity'
thf(fact_5137_rat__uminus__code,axiom,
! [P2: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A8: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A8 ) )
@ ( quotient_of @ P2 ) ) ) ).
% rat_uminus_code
thf(fact_5138_rat__abs__code,axiom,
! [P2: rat] :
( ( quotient_of @ ( abs_abs @ rat @ P2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A8: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A8 ) )
@ ( quotient_of @ P2 ) ) ) ).
% rat_abs_code
thf(fact_5139_normalize__denom__pos,axiom,
! [R2: product_prod @ int @ int,P2: int,Q3: int] :
( ( ( normalize @ R2 )
= ( product_Pair @ int @ int @ P2 @ Q3 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% normalize_denom_pos
thf(fact_5140_normalize__crossproduct,axiom,
! [Q3: int,S: int,P2: int,R2: int] :
( ( Q3
!= ( zero_zero @ int ) )
=> ( ( S
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P2 @ Q3 ) )
= ( normalize @ ( product_Pair @ int @ int @ R2 @ S ) ) )
=> ( ( times_times @ int @ P2 @ S )
= ( times_times @ int @ R2 @ Q3 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_5141_rat__sgn__code,axiom,
! [P2: rat] :
( ( quotient_of @ ( sgn_sgn @ rat @ P2 ) )
= ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P2 ) ) ) @ ( one_one @ int ) ) ) ).
% rat_sgn_code
thf(fact_5142_filterlim__power__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [F3: A > B,F6: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F6 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N )
@ ( at_infinity @ B )
@ F6 ) ) ) ) ).
% filterlim_power_at_infinity
thf(fact_5143_quotient__of__int,axiom,
! [A3: int] :
( ( quotient_of @ ( of_int @ A3 ) )
= ( product_Pair @ int @ int @ A3 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_5144_Gcd__eq__Max,axiom,
! [M7: set @ nat] :
( ( finite_finite2 @ nat @ M7 )
=> ( ( M7
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( gcd_Gcd @ nat @ M7 )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image2 @ nat @ ( set @ nat )
@ ^ [M5: nat] :
( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M5 ) )
@ M7 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_5145_polyfun__extremal,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [C3: nat > A,K2: nat,N: nat,B5: real] :
( ( ( C3 @ K2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K2 )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( eventually @ A
@ ^ [Z4: A] :
( ord_less_eq @ real @ B5
@ ( 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 ) ) ) )
@ ( at_infinity @ A ) ) ) ) ) ) ).
% polyfun_extremal
thf(fact_5146_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_5147_eventually__sequentially__seg,axiom,
! [P: nat > $o,K2: nat] :
( ( eventually @ nat
@ ^ [N3: nat] : ( P @ ( plus_plus @ nat @ N3 @ K2 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_5148_eventually__top,axiom,
! [A: $tType,P: A > $o] :
( ( eventually @ A @ P @ ( top_top @ ( filter @ A ) ) )
= ( ! [X8: A] : ( P @ X8 ) ) ) ).
% eventually_top
thf(fact_5149_eventually__const,axiom,
! [A: $tType,F6: filter @ A,P: $o] :
( ( F6
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A
@ ^ [X4: A] : P
@ F6 )
= P ) ) ).
% eventually_const
thf(fact_5150_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A] :
( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% Max_singleton
thf(fact_5151_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X3 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ X4 @ X3 ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_5152_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X3 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less @ A @ X4 @ X3 ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_5153_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ B,C3: A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [Uu3: B] : C3
@ A6 ) )
= C3 ) ) ) ) ).
% Max_const
thf(fact_5154_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ) ).
% Max_insert
thf(fact_5155_eventually__Sup,axiom,
! [A: $tType,P: A > $o,S3: set @ ( filter @ A )] :
( ( eventually @ A @ P @ ( complete_Sup_Sup @ ( filter @ A ) @ S3 ) )
= ( ! [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ S3 )
=> ( eventually @ A @ P @ X4 ) ) ) ) ).
% eventually_Sup
thf(fact_5156_eventually__at__bot__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [C3: A] :
( eventually @ A
@ ^ [X4: A] : X4 != C3
@ ( at_bot @ A ) ) ) ).
% eventually_at_bot_not_equal
thf(fact_5157_filterlim__iff,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F4: A > B,F26: filter @ B,F15: filter @ A] :
! [P4: B > $o] :
( ( eventually @ B @ P4 @ F26 )
=> ( eventually @ A
@ ^ [X4: A] : ( P4 @ ( F4 @ X4 ) )
@ F15 ) ) ) ) ).
% filterlim_iff
thf(fact_5158_filterlim__cong,axiom,
! [A: $tType,B: $tType,F13: filter @ A,F14: filter @ A,F24: filter @ B,F25: filter @ B,F3: B > A,G3: B > A] :
( ( F13 = F14 )
=> ( ( F24 = F25 )
=> ( ( eventually @ B
@ ^ [X4: B] :
( ( F3 @ X4 )
= ( G3 @ X4 ) )
@ F24 )
=> ( ( filterlim @ B @ A @ F3 @ F13 @ F24 )
= ( filterlim @ B @ A @ G3 @ F14 @ F25 ) ) ) ) ) ).
% filterlim_cong
thf(fact_5159_eventually__compose__filterlim,axiom,
! [A: $tType,B: $tType,P: A > $o,F6: filter @ A,F3: B > A,G7: filter @ B] :
( ( eventually @ A @ P @ F6 )
=> ( ( filterlim @ B @ A @ F3 @ F6 @ G7 )
=> ( eventually @ B
@ ^ [X4: B] : ( P @ ( F3 @ X4 ) )
@ G7 ) ) ) ).
% eventually_compose_filterlim
thf(fact_5160_trivial__limit__def,axiom,
! [A: $tType,F6: filter @ A] :
( ( F6
= ( bot_bot @ ( filter @ A ) ) )
= ( eventually @ A
@ ^ [X4: A] : $false
@ F6 ) ) ).
% trivial_limit_def
thf(fact_5161_eventually__const__iff,axiom,
! [A: $tType,P: $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] : P
@ F6 )
= ( P
| ( F6
= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% eventually_const_iff
thf(fact_5162_False__imp__not__eventually,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ! [X5: A] :
~ ( P @ X5 )
=> ( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ~ ( eventually @ A @ P @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_5163_eventually__happens_H,axiom,
! [A: $tType,F6: filter @ A,P: A > $o] :
( ( F6
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A @ P @ F6 )
=> ? [X_12: A] : ( P @ X_12 ) ) ) ).
% eventually_happens'
thf(fact_5164_eventually__happens,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ( eventually @ A @ P @ Net )
=> ( ( Net
= ( bot_bot @ ( filter @ A ) ) )
| ? [X_12: A] : ( P @ X_12 ) ) ) ).
% eventually_happens
thf(fact_5165_eventually__bot,axiom,
! [A: $tType,P: A > $o] : ( eventually @ A @ P @ ( bot_bot @ ( filter @ A ) ) ) ).
% eventually_bot
thf(fact_5166_not__eventually__impI,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F6 )
=> ( ~ ( eventually @ A @ Q @ F6 )
=> ~ ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
@ F6 ) ) ) ).
% not_eventually_impI
thf(fact_5167_eventually__conj__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) )
@ F6 )
= ( ( eventually @ A @ P @ F6 )
& ( eventually @ A @ Q @ F6 ) ) ) ).
% eventually_conj_iff
thf(fact_5168_eventually__rev__mp,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F6 )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
@ F6 )
=> ( eventually @ A @ Q @ F6 ) ) ) ).
% eventually_rev_mp
thf(fact_5169_eventually__subst,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [N3: A] :
( ( P @ N3 )
= ( Q @ N3 ) )
@ F6 )
=> ( ( eventually @ A @ P @ F6 )
= ( eventually @ A @ Q @ F6 ) ) ) ).
% eventually_subst
thf(fact_5170_eventually__elim2,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o,R: A > $o] :
( ( eventually @ A @ P @ F6 )
=> ( ( eventually @ A @ Q @ F6 )
=> ( ! [I3: A] :
( ( P @ I3 )
=> ( ( Q @ I3 )
=> ( R @ I3 ) ) )
=> ( eventually @ A @ R @ F6 ) ) ) ) ).
% eventually_elim2
thf(fact_5171_eventually__conj,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F6 )
=> ( ( eventually @ A @ Q @ F6 )
=> ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) )
@ F6 ) ) ) ).
% eventually_conj
thf(fact_5172_eventually__True,axiom,
! [A: $tType,F6: filter @ A] :
( eventually @ A
@ ^ [X4: A] : $true
@ F6 ) ).
% eventually_True
thf(fact_5173_eventually__mp,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
@ F6 )
=> ( ( eventually @ A @ P @ F6 )
=> ( eventually @ A @ Q @ F6 ) ) ) ).
% eventually_mp
thf(fact_5174_eventually__frequently__const__simps_I3_J,axiom,
! [A: $tType,P: A > $o,C4: $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
| C4 )
@ F6 )
= ( ( eventually @ A @ P @ F6 )
| C4 ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_5175_eventually__frequently__const__simps_I4_J,axiom,
! [A: $tType,C4: $o,P: A > $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( C4
| ( P @ X4 ) )
@ F6 )
= ( C4
| ( eventually @ A @ P @ F6 ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_5176_eventually__frequently__const__simps_I6_J,axiom,
! [A: $tType,C4: $o,P: A > $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( C4
=> ( P @ X4 ) )
@ F6 )
= ( C4
=> ( eventually @ A @ P @ F6 ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_5177_always__eventually,axiom,
! [A: $tType,P: A > $o,F6: filter @ A] :
( ! [X_12: A] : ( P @ X_12 )
=> ( eventually @ A @ P @ F6 ) ) ).
% always_eventually
thf(fact_5178_not__eventuallyD,axiom,
! [A: $tType,P: A > $o,F6: filter @ A] :
( ~ ( eventually @ A @ P @ F6 )
=> ? [X5: A] :
~ ( P @ X5 ) ) ).
% not_eventuallyD
thf(fact_5179_eventually__mono,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F6 )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( eventually @ A @ Q @ F6 ) ) ) ).
% eventually_mono
thf(fact_5180_filter__eq__iff,axiom,
! [A: $tType] :
( ( ^ [Y5: filter @ A,Z: filter @ A] : Y5 = Z )
= ( ^ [F9: filter @ A,F10: filter @ A] :
! [P4: A > $o] :
( ( eventually @ A @ P4 @ F9 )
= ( eventually @ A @ P4 @ F10 ) ) ) ) ).
% filter_eq_iff
thf(fact_5181_eventuallyI,axiom,
! [A: $tType,P: A > $o,F6: filter @ A] :
( ! [X_12: A] : ( P @ X_12 )
=> ( eventually @ A @ P @ F6 ) ) ).
% eventuallyI
thf(fact_5182_eventually__inf,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,F11: filter @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F6 @ F11 ) )
= ( ? [Q6: A > $o,R6: A > $o] :
( ( eventually @ A @ Q6 @ F6 )
& ( eventually @ A @ R6 @ F11 )
& ! [X4: A] :
( ( ( Q6 @ X4 )
& ( R6 @ X4 ) )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_inf
thf(fact_5183_filter__leD,axiom,
! [A: $tType,F6: filter @ A,F11: filter @ A,P: A > $o] :
( ( ord_less_eq @ ( filter @ A ) @ F6 @ F11 )
=> ( ( eventually @ A @ P @ F11 )
=> ( eventually @ A @ P @ F6 ) ) ) ).
% filter_leD
thf(fact_5184_filter__leI,axiom,
! [A: $tType,F11: filter @ A,F6: filter @ A] :
( ! [P7: A > $o] :
( ( eventually @ A @ P7 @ F11 )
=> ( eventually @ A @ P7 @ F6 ) )
=> ( ord_less_eq @ ( filter @ A ) @ F6 @ F11 ) ) ).
% filter_leI
thf(fact_5185_le__filter__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( filter @ A ) )
= ( ^ [F9: filter @ A,F10: filter @ A] :
! [P4: A > $o] :
( ( eventually @ A @ P4 @ F10 )
=> ( eventually @ A @ P4 @ F9 ) ) ) ) ).
% le_filter_def
thf(fact_5186_eventually__sup,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,F11: filter @ A] :
( ( eventually @ A @ P @ ( sup_sup @ ( filter @ A ) @ F6 @ F11 ) )
= ( ( eventually @ A @ P @ F6 )
& ( eventually @ A @ P @ F11 ) ) ) ).
% eventually_sup
thf(fact_5187_eventually__at__top__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C3: A] :
( eventually @ A
@ ^ [X4: A] : X4 != C3
@ ( at_top @ A ) ) ) ).
% eventually_at_top_not_equal
thf(fact_5188_eventually__False__sequentially,axiom,
~ ( eventually @ nat
@ ^ [N3: nat] : $false
@ ( at_top @ nat ) ) ).
% eventually_False_sequentially
thf(fact_5189_eventually__principal,axiom,
! [A: $tType,P: A > $o,S3: set @ A] :
( ( eventually @ A @ P @ ( principal @ A @ S3 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ S3 )
=> ( P @ X4 ) ) ) ) ).
% eventually_principal
thf(fact_5190_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,P: A > $o] :
( ! [X5: A] :
( ( ord_less_eq @ A @ C3 @ X5 )
=> ( P @ X5 ) )
=> ( eventually @ A @ P @ ( at_top @ A ) ) ) ) ).
% eventually_at_top_linorderI
thf(fact_5191_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N6: A] :
! [N3: A] :
( ( ord_less_eq @ A @ N6 @ N3 )
=> ( P @ N3 ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_5192_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N6: A] :
! [N3: A] :
( ( ord_less @ A @ N6 @ N3 )
=> ( P @ N3 ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_5193_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
= ( ? [N6: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( P @ N3 ) ) ) ) ).
% eventually_sequentially
thf(fact_5194_eventually__sequentiallyI,axiom,
! [C3: nat,P: nat > $o] :
( ! [X5: nat] :
( ( ord_less_eq @ nat @ C3 @ X5 )
=> ( P @ X5 ) )
=> ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentiallyI
thf(fact_5195_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N6: A] :
! [N3: A] :
( ( ord_less_eq @ A @ N3 @ N6 )
=> ( P @ N3 ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_5196_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N6: A] :
! [N3: A] :
( ( ord_less @ A @ N3 @ N6 )
=> ( P @ N3 ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_5197_Max__ge,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ).
% Max_ge
thf(fact_5198_Max__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ A6 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( lattic643756798349783984er_Max @ A @ A6 )
= X3 ) ) ) ) ) ).
% Max_eqI
thf(fact_5199_Max__eq__if,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ B5 )
& ( ord_less_eq @ A @ X5 @ Xa ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B5 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ A6 )
& ( ord_less_eq @ A @ X5 @ Xa ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A6 )
= ( lattic643756798349783984er_Max @ A @ B5 ) ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_5200_Max_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ord_less_eq @ A @ A3 @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ).
% Max.coboundedI
thf(fact_5201_Max__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ A6 ) ) ) ) ).
% Max_in
thf(fact_5202_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_5203_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_5204_le__sequentially,axiom,
! [F6: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F6 @ ( at_top @ nat ) )
= ( ! [N6: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N6 ) @ F6 ) ) ) ).
% le_sequentially
thf(fact_5205_sequentially__offset,axiom,
! [P: nat > $o,K2: nat] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [I4: nat] : ( P @ ( plus_plus @ nat @ I4 @ K2 ) )
@ ( at_top @ nat ) ) ) ).
% sequentially_offset
thf(fact_5206_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ A @ X4 @ C3 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_5207_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C3: A] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ A @ X4 @ C3 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_5208_filterlim__principal,axiom,
! [B: $tType,A: $tType,F3: A > B,S3: set @ B,F6: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( principal @ B @ S3 ) @ F6 )
= ( eventually @ A
@ ^ [X4: A] : ( member @ B @ ( F3 @ X4 ) @ S3 )
@ F6 ) ) ).
% filterlim_principal
thf(fact_5209_filterlim__mono__eventually,axiom,
! [B: $tType,A: $tType,F3: A > B,F6: filter @ B,G7: filter @ A,F11: filter @ B,G8: filter @ A,F8: A > B] :
( ( filterlim @ A @ B @ F3 @ F6 @ G7 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F6 @ F11 )
=> ( ( ord_less_eq @ ( filter @ A ) @ G8 @ G7 )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
= ( F8 @ X4 ) )
@ G8 )
=> ( filterlim @ A @ B @ F8 @ F11 @ G8 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_5210_le__principal,axiom,
! [A: $tType,F6: filter @ A,A6: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F6 @ ( principal @ A @ A6 ) )
= ( eventually @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A6 )
@ F6 ) ) ).
% le_principal
thf(fact_5211_eventually__INF1,axiom,
! [B: $tType,A: $tType,I: A,I5: set @ A,P: B > $o,F6: A > ( filter @ B )] :
( ( member @ A @ I @ I5 )
=> ( ( eventually @ B @ P @ ( F6 @ I ) )
=> ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F6 @ I5 ) ) ) ) ) ).
% eventually_INF1
thf(fact_5212_eventually__inf__principal,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,S: set @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F6 @ ( principal @ A @ S ) ) )
= ( eventually @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S )
=> ( P @ X4 ) )
@ F6 ) ) ).
% eventually_inf_principal
thf(fact_5213_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P: B > $o,G3: B > A] :
( ! [X5: A,Y4: A] :
( ( Q @ X5 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) ) ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( ( F3 @ ( G3 @ X5 ) )
= X5 ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( Q @ ( G3 @ X5 ) ) )
=> ( ( 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_5214_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ( ord_less_eq @ A @ A5 @ X3 ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X3 ) ) ) ) ) ).
% Max.boundedI
thf(fact_5215_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X3 )
=> ! [A18: A] :
( ( member @ A @ A18 @ A6 )
=> ( ord_less_eq @ A @ A18 @ X3 ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_5216_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,M2: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M2
= ( lattic643756798349783984er_Max @ A @ A6 ) )
= ( ( member @ A @ M2 @ A6 )
& ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ X4 @ M2 ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_5217_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A6 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ord_less_eq @ A @ X3 @ X4 ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_5218_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,M2: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A6 )
= M2 )
= ( ( member @ A @ M2 @ A6 )
& ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ X4 @ M2 ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_5219_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A6 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ord_less @ A @ X3 @ X4 ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_5220_tendsto__sandwich,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,G3: B > A,Net: filter @ B,H: B > A,C3: A] :
( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G3 @ N3 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ A @ ( G3 @ N3 ) @ ( H @ N3 ) )
@ Net )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ Net )
=> ( ( filterlim @ B @ A @ H @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ Net )
=> ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ Net ) ) ) ) ) ) ).
% tendsto_sandwich
thf(fact_5221_Max__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ A6 )
=> ( ord_less_eq @ A @ B4 @ A3 ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ A3 @ A6 ) )
= A3 ) ) ) ) ).
% Max_insert2
thf(fact_5222_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,F6: filter @ B,G3: B > A] :
( ( filterlim @ B @ A @ F3 @ ( at_top @ A ) @ F6 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ F6 )
=> ( filterlim @ B @ A @ G3 @ ( at_top @ A ) @ F6 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_5223_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F6: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F6 )
= ( ! [Z8: B] :
( ( ord_less_eq @ B @ C3 @ Z8 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z8 @ ( F3 @ X4 ) )
@ F6 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_5224_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F6: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F6 )
= ( ! [Z8: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z8 @ ( F3 @ X4 ) )
@ F6 ) ) ) ) ).
% filterlim_at_top
thf(fact_5225_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F6: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F6 )
= ( ! [Z8: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ Z8 @ ( F3 @ X4 ) )
@ F6 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_5226_cSup__eq__Max,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= ( lattic643756798349783984er_Max @ A @ X6 ) ) ) ) ) ).
% cSup_eq_Max
thf(fact_5227_Max__Sup,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A6 )
= ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% Max_Sup
thf(fact_5228_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F6: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F6 )
= ( ! [Z8: B] :
( ( ord_less_eq @ B @ Z8 @ C3 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z8 )
@ F6 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_5229_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F6: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F6 )
= ( ! [Z8: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z8 )
@ F6 ) ) ) ) ).
% filterlim_at_bot
thf(fact_5230_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F3: A > B,F6: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F6 )
= ( ! [Z8: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ ( F3 @ X4 ) @ Z8 )
@ F6 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_5231_countable__basis__at__decseq,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X3: A] :
~ ! [A10: nat > ( set @ A )] :
( ! [I2: nat] : ( topolo1002775350975398744n_open @ A @ ( A10 @ I2 ) )
=> ( ! [I2: nat] : ( member @ A @ X3 @ ( A10 @ I2 ) )
=> ~ ! [S9: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S9 )
=> ( ( member @ A @ X3 @ S9 )
=> ( eventually @ nat
@ ^ [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( A10 @ I4 ) @ S9 )
@ ( at_top @ nat ) ) ) ) ) ) ) ).
% countable_basis_at_decseq
thf(fact_5232_Max_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( lattic643756798349783984er_Max @ A @ A6 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Max.infinite
thf(fact_5233_eventually__Inf__base,axiom,
! [A: $tType,B5: set @ ( filter @ A ),P: A > $o] :
( ( B5
!= ( bot_bot @ ( set @ ( filter @ A ) ) ) )
=> ( ! [F5: filter @ A] :
( ( member @ ( filter @ A ) @ F5 @ B5 )
=> ! [G5: filter @ A] :
( ( member @ ( filter @ A ) @ G5 @ B5 )
=> ? [X: filter @ A] :
( ( member @ ( filter @ A ) @ X @ B5 )
& ( ord_less_eq @ ( filter @ A ) @ X @ ( inf_inf @ ( filter @ A ) @ F5 @ G5 ) ) ) ) )
=> ( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B5 ) )
= ( ? [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ B5 )
& ( eventually @ A @ P @ X4 ) ) ) ) ) ) ).
% eventually_Inf_base
thf(fact_5234_eventually__INF__finite,axiom,
! [B: $tType,A: $tType,A6: set @ A,P: B > $o,F6: A > ( filter @ B )] :
( ( finite_finite2 @ A @ A6 )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F6 @ A6 ) ) )
= ( ? [Q6: A > B > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( eventually @ B @ ( Q6 @ X4 ) @ ( F6 @ X4 ) ) )
& ! [Y3: B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( Q6 @ X4 @ Y3 ) )
=> ( P @ Y3 ) ) ) ) ) ) ).
% eventually_INF_finite
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
@ ^ [X4: A] : ( P @ ( plus_plus @ A @ X4 @ A3 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% eventually_at_to_0
thf(fact_5236_decreasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [L: A,F3: B > A,F6: filter @ B] :
( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ A @ L @ ( F3 @ N3 ) )
@ F6 )
=> ( ! [X5: A] :
( ( ord_less @ A @ L @ X5 )
=> ( eventually @ B
@ ^ [N3: B] : ( ord_less @ A @ ( F3 @ N3 ) @ X5 )
@ F6 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F6 ) ) ) ) ).
% decreasing_tendsto
thf(fact_5237_increasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,L: A,F6: filter @ B] :
( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ L )
@ F6 )
=> ( ! [X5: A] :
( ( ord_less @ A @ X5 @ L )
=> ( eventually @ B
@ ^ [N3: B] : ( ord_less @ A @ X5 @ ( F3 @ N3 ) )
@ F6 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F6 ) ) ) ) ).
% increasing_tendsto
thf(fact_5238_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F6: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F6 )
= ( ! [Z8: B] :
( ( ord_less @ B @ C3 @ Z8 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z8 @ ( F3 @ X4 ) )
@ F6 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_5239_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F6: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F6 )
= ( ! [Z8: B] :
( ( ord_less @ B @ Z8 @ C3 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z8 )
@ F6 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_5240_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M7: set @ A,N5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M7 @ N5 )
=> ( ( M7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N5 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M7 ) @ ( lattic643756798349783984er_Max @ A @ N5 ) ) ) ) ) ) ).
% Max_mono
thf(fact_5241_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ ( lattic643756798349783984er_Max @ A @ B5 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_5242_tendsto__upperbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: B > A,X3: A,F6: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ F6 )
=> ( ( eventually @ B
@ ^ [I4: B] : ( ord_less_eq @ A @ ( F3 @ I4 ) @ A3 )
@ F6 )
=> ( ( F6
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ X3 @ A3 ) ) ) ) ) ).
% tendsto_upperbound
thf(fact_5243_tendsto__lowerbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: B > A,X3: A,F6: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ F6 )
=> ( ( eventually @ B
@ ^ [I4: B] : ( ord_less_eq @ A @ A3 @ ( F3 @ I4 ) )
@ F6 )
=> ( ( F6
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ A3 @ X3 ) ) ) ) ) ).
% tendsto_lowerbound
thf(fact_5244_tendsto__le,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F6: filter @ B,F3: B > A,X3: A,G3: B > A,Y: A] :
( ( F6
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ F6 )
=> ( ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F6 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( ord_less_eq @ A @ ( G3 @ X4 ) @ ( F3 @ X4 ) )
@ F6 )
=> ( ord_less_eq @ A @ Y @ X3 ) ) ) ) ) ) ).
% tendsto_le
thf(fact_5245_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H: A > A,N5: set @ A] :
( ! [X5: A,Y4: A] :
( ( H @ ( ord_max @ A @ X5 @ Y4 ) )
= ( ord_max @ A @ ( H @ X5 ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite2 @ A @ N5 )
=> ( ( N5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic643756798349783984er_Max @ A @ N5 ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ H @ N5 ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_5246_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B5 ) @ ( lattic643756798349783984er_Max @ A @ A6 ) )
= ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ) ).
% Max.subset
thf(fact_5247_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X3 @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_5248_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y4: A] : ( member @ A @ ( ord_max @ A @ X5 @ Y4 ) @ ( insert2 @ A @ X5 @ ( insert2 @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ A6 ) ) ) ) ) ).
% Max.closed
thf(fact_5249_Max_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ ( lattic643756798349783984er_Max @ A @ B5 ) ) ) ) ) ) ) ) ).
% Max.union
thf(fact_5250_eventually__INF,axiom,
! [A: $tType,B: $tType,P: A > $o,F6: B > ( filter @ A ),B5: set @ B] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F6 @ B5 ) ) )
= ( ? [X8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ X8 @ B5 )
& ( finite_finite2 @ B @ X8 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F6 @ X8 ) ) ) ) ) ) ).
% eventually_INF
thf(fact_5251_Max_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( finite_fold @ A @ A @ ( ord_max @ A ) @ X3 @ A6 ) ) ) ) ).
% Max.eq_fold
thf(fact_5252_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_5253_card__le__Suc__Max,axiom,
! [S3: set @ nat] :
( ( finite_finite2 @ nat @ S3 )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ S3 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S3 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_5254_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M5: nat,N3: nat] :
( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ N3 ) @ M5 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_5255_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S3: set @ B,F3: B > A,K2: A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F3 @ X4 ) @ K2 )
@ S3 ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image2 @ B @ A @ F3 @ S3 ) ) @ K2 ) ) ) ) ) ).
% Max_add_commute
thf(fact_5256_eventually__Inf,axiom,
! [A: $tType,P: A > $o,B5: set @ ( filter @ A )] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B5 ) )
= ( ? [X8: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X8 @ B5 )
& ( finite_finite2 @ ( filter @ A ) @ X8 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ X8 ) ) ) ) ) ).
% eventually_Inf
thf(fact_5257_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A6 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A6 )
= ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_5258_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X3 @ A6 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_5259_filterlim__at__top__at__left,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P: B > $o,G3: B > A,A3: A] :
( ! [X5: A,Y4: A] :
( ( Q @ X5 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) ) ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( ( F3 @ ( G3 @ X5 ) )
= X5 ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( Q @ ( G3 @ X5 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) )
=> ( ! [B4: A] :
( ( Q @ B4 )
=> ( ord_less @ A @ B4 @ A3 ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_left
thf(fact_5260_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B5: set @ A,F6: A > ( filter @ B ),P: B > $o] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ B5 )
=> ! [B4: A] :
( ( member @ A @ B4 @ B5 )
=> ? [X: A] :
( ( member @ A @ X @ B5 )
& ( ord_less_eq @ ( filter @ B ) @ ( F6 @ X ) @ ( inf_inf @ ( filter @ B ) @ ( F6 @ A5 ) @ ( F6 @ B4 ) ) ) ) ) )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F6 @ B5 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ B5 )
& ( eventually @ B @ P @ ( F6 @ X4 ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_5261_filterlim__at__bot__at__right,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P: B > $o,G3: B > A,A3: A] :
( ! [X5: A,Y4: A] :
( ( Q @ X5 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) ) ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( ( F3 @ ( G3 @ X5 ) )
= X5 ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( Q @ ( G3 @ X5 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( ! [B4: A] :
( ( Q @ B4 )
=> ( ord_less @ A @ A3 @ B4 ) )
=> ( ( eventually @ B @ P @ ( at_bot @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_bot_at_right
thf(fact_5262_filterlim__at__withinI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,C3: A,F6: filter @ B,A6: set @ A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F6 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( member @ A @ ( F3 @ X4 ) @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ F6 )
=> ( filterlim @ B @ A @ F3 @ ( topolo174197925503356063within @ A @ C3 @ A6 ) @ F6 ) ) ) ) ).
% filterlim_at_withinI
thf(fact_5263_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( ord_max @ A @ X4 ) @ Y3 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_5264_sum__le__card__Max,axiom,
! [A: $tType,A6: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A6 ) @ ( times_times @ nat @ ( finite_card @ A @ A6 ) @ ( lattic643756798349783984er_Max @ nat @ ( image2 @ A @ nat @ F3 @ A6 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_5265_Frct__code__post_I5_J,axiom,
! [K2: num] :
( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ K2 ) ) )
= ( divide_divide @ rat @ ( one_one @ rat ) @ ( numeral_numeral @ rat @ K2 ) ) ) ).
% Frct_code_post(5)
thf(fact_5266_summable__Cauchy_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G3: nat > real] :
( ( eventually @ nat
@ ^ [M5: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M5 @ N3 ) ) ) @ ( G3 @ M5 ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_Cauchy'
thf(fact_5267_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_5268_gcd__add2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M2: A,N: A] :
( ( gcd_gcd @ A @ M2 @ ( plus_plus @ A @ M2 @ N ) )
= ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% gcd_add2
thf(fact_5269_gcd__add1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M2: A,N: A] :
( ( gcd_gcd @ A @ ( plus_plus @ A @ M2 @ N ) @ N )
= ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% gcd_add1
thf(fact_5270_gcd__exp,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( gcd_gcd @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( power_power @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ N ) ) ) ).
% gcd_exp
thf(fact_5271_gcd__neg__numeral__2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,N: num] :
( ( gcd_gcd @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( gcd_gcd @ A @ A3 @ ( numeral_numeral @ A @ N ) ) ) ) ).
% gcd_neg_numeral_2
thf(fact_5272_gcd__neg__numeral__1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N: num,A3: A] :
( ( gcd_gcd @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ A3 )
= ( gcd_gcd @ A @ ( numeral_numeral @ A @ N ) @ A3 ) ) ) ).
% gcd_neg_numeral_1
thf(fact_5273_Gcd__insert,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,A6: set @ A] :
( ( gcd_Gcd @ A @ ( insert2 @ A @ A3 @ A6 ) )
= ( gcd_gcd @ A @ A3 @ ( gcd_Gcd @ A @ A6 ) ) ) ) ).
% Gcd_insert
thf(fact_5274_gcd__neg__numeral__2__int,axiom,
! [X3: int,N: num] :
( ( gcd_gcd @ int @ X3 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( gcd_gcd @ int @ X3 @ ( numeral_numeral @ int @ N ) ) ) ).
% gcd_neg_numeral_2_int
thf(fact_5275_gcd__neg__numeral__1__int,axiom,
! [N: num,X3: int] :
( ( gcd_gcd @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ X3 )
= ( gcd_gcd @ int @ ( numeral_numeral @ int @ N ) @ X3 ) ) ).
% gcd_neg_numeral_1_int
thf(fact_5276_Gcd__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_Gcd @ A @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% Gcd_2
thf(fact_5277_eventually__all__finite,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [P: A > B > $o,Net: filter @ A] :
( ! [Y4: B] :
( eventually @ A
@ ^ [X4: A] : ( P @ X4 @ Y4 )
@ Net )
=> ( eventually @ A
@ ^ [X4: A] :
! [X8: B] : ( P @ X4 @ X8 )
@ Net ) ) ) ).
% eventually_all_finite
thf(fact_5278_gcd__add__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M2: A,K2: A,N: A] :
( ( gcd_gcd @ A @ M2 @ ( plus_plus @ A @ ( times_times @ A @ K2 @ M2 ) @ N ) )
= ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% gcd_add_mult
thf(fact_5279_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X4: A] :
! [Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( P @ Y3 ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_5280_Frct__code__post_I2_J,axiom,
! [A3: int] :
( ( frct @ ( product_Pair @ int @ int @ A3 @ ( zero_zero @ int ) ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(2)
thf(fact_5281_Frct__code__post_I1_J,axiom,
! [A3: int] :
( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A3 ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(1)
thf(fact_5282_Frct__code__post_I7_J,axiom,
! [A3: int,B2: int] :
( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A3 ) @ B2 ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% Frct_code_post(7)
thf(fact_5283_Frct__code__post_I8_J,axiom,
! [A3: int,B2: int] :
( ( frct @ ( product_Pair @ int @ int @ A3 @ ( uminus_uminus @ int @ B2 ) ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% Frct_code_post(8)
thf(fact_5284_Frct__code__post_I3_J,axiom,
( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) )
= ( one_one @ rat ) ) ).
% Frct_code_post(3)
thf(fact_5285_Frct__code__post_I4_J,axiom,
! [K2: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K2 ) @ ( one_one @ int ) ) )
= ( numeral_numeral @ rat @ K2 ) ) ).
% Frct_code_post(4)
thf(fact_5286_Frct__code__post_I6_J,axiom,
! [K2: num,L: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K2 ) @ ( numeral_numeral @ int @ L ) ) )
= ( divide_divide @ rat @ ( numeral_numeral @ rat @ K2 ) @ ( numeral_numeral @ rat @ L ) ) ) ).
% Frct_code_post(6)
thf(fact_5287_summable__bounded__partials,axiom,
! [A: $tType] :
( ( ( real_V8037385150606011577_space @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: nat > A,G3: nat > real] :
( ( eventually @ nat
@ ^ [X02: nat] :
! [A8: nat] :
( ( ord_less_eq @ nat @ X02 @ A8 )
=> ! [B8: nat] :
( ( ord_less @ nat @ A8 @ B8 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or3652927894154168847AtMost @ nat @ A8 @ B8 ) ) ) @ ( G3 @ A8 ) ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_bounded_partials
thf(fact_5288_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P4: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P4 @ X4 )
& ! [Y3: A] :
( ( P4 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_5289_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
= Y )
=> ( ( ? [Uu: $o,Uv: $o] :
( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( Y
= ( Xa2
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( Y
= ( ~ ( ( Deg2 = Xa2 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ 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 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_5290_gcd__Suc__0,axiom,
! [M2: nat] :
( ( gcd_gcd @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% gcd_Suc_0
thf(fact_5291_ball__empty,axiom,
! [A: $tType,P: A > $o,X: A] :
( ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X ) ) ).
% ball_empty
thf(fact_5292_ball__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) )
=> ( P @ X4 ) ) )
= ( ! [X8: A] : ( P @ X8 ) ) ) ).
% ball_UNIV
thf(fact_5293_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_5294_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K2: A] :
( ( ord_less_eq @ A @ L @ K2 )
=> ( ( set_or3652927894154168847AtMost @ A @ K2 @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanAtMost_empty
thf(fact_5295_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K2: A,L: A] :
( ( ( set_or3652927894154168847AtMost @ A @ K2 @ L )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ K2 @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_5296_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K2: A,L: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or3652927894154168847AtMost @ A @ K2 @ L ) )
= ( ~ ( ord_less @ A @ K2 @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_5297_image__add__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ C3 ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_5298_eventually__ex,axiom,
! [B: $tType,A: $tType,P: A > B > $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
? [X8: B] : ( P @ X4 @ X8 )
@ F6 )
= ( ? [Y10: A > B] :
( eventually @ A
@ ^ [X4: A] : ( P @ X4 @ ( Y10 @ X4 ) )
@ F6 ) ) ) ).
% eventually_ex
thf(fact_5299_Ball__Collect,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A7: set @ A,P4: A > $o] : ( ord_less_eq @ ( set @ A ) @ A7 @ ( collect @ A @ P4 ) ) ) ) ).
% Ball_Collect
thf(fact_5300_Ioc__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ( set_or3652927894154168847AtMost @ A @ A3 @ B2 )
= ( set_or3652927894154168847AtMost @ A @ C3 @ D3 ) )
= ( ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ D3 @ C3 ) )
| ( ( A3 = C3 )
& ( B2 = D3 ) ) ) ) ) ).
% Ioc_inj
thf(fact_5301_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X4: B] :
( ( Uu3
= ( F3 @ X4 ) )
& ( member @ B @ X4 @ A6 ) ) )
= ( image2 @ B @ A @ F3 @ A6 ) ) ).
% Setcompr_eq_image
thf(fact_5302_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F3: B > A,P: B > $o] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X4: B] :
( ( Uu3
= ( F3 @ X4 ) )
& ( P @ X4 ) ) )
= ( image2 @ B @ A @ F3 @ ( collect @ B @ P ) ) ) ).
% setcompr_eq_image
thf(fact_5303_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A7: set @ A,P4: A > $o] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( P4 @ X4 ) ) ) ) ).
% Ball_def
thf(fact_5304_open__diagonal__complement,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y3: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y3 ) )
& ( X4 != Y3 ) ) ) ) ) ).
% open_diagonal_complement
thf(fact_5305_gcd__le1__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ A3 ) ) ).
% gcd_le1_nat
thf(fact_5306_gcd__le2__nat,axiom,
! [B2: nat,A3: nat] :
( ( B2
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ B2 ) ) ).
% gcd_le2_nat
thf(fact_5307_gcd__diff2__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ N @ M2 ) @ N )
= ( gcd_gcd @ nat @ M2 @ N ) ) ) ).
% gcd_diff2_nat
thf(fact_5308_gcd__diff1__nat,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N )
= ( gcd_gcd @ nat @ M2 @ N ) ) ) ).
% gcd_diff1_nat
thf(fact_5309_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U )
= ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_5310_Ioc__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D3 ) )
= ( ( ord_less_eq @ A @ B2 @ A3 )
| ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% Ioc_subset_iff
thf(fact_5311_ivl__disj__un__two_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_5312_ivl__disj__int__two_I6_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(6)
thf(fact_5313_open__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y3: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y3 ) )
& ( ord_less @ A @ X4 @ Y3 ) ) ) ) ) ).
% open_subdiagonal
thf(fact_5314_open__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y3: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y3 ) )
& ( ord_less @ A @ Y3 @ X4 ) ) ) ) ) ).
% open_superdiagonal
thf(fact_5315_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F3: B > A] :
( ( collect @ A
@ ^ [U2: A] :
? [X4: B] :
( U2
= ( F3 @ X4 ) ) )
= ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% full_SetCompr_eq
thf(fact_5316_eventually__ball__finite,axiom,
! [A: $tType,B: $tType,A6: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( eventually @ B
@ ^ [Y3: B] : ( P @ Y3 @ X5 )
@ Net ) )
=> ( eventually @ B
@ ^ [X4: B] :
! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( P @ X4 @ Y3 ) )
@ Net ) ) ) ).
% eventually_ball_finite
thf(fact_5317_eventually__ball__finite__distrib,axiom,
! [B: $tType,A: $tType,A6: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( eventually @ B
@ ^ [X4: B] :
! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( P @ X4 @ Y3 ) )
@ Net )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( eventually @ B
@ ^ [Y3: B] : ( P @ Y3 @ X4 )
@ Net ) ) ) ) ) ).
% eventually_ball_finite_distrib
thf(fact_5318_Gcd__in,axiom,
! [A6: set @ nat] :
( ! [A5: nat,B4: nat] :
( ( member @ nat @ A5 @ A6 )
=> ( ( member @ nat @ B4 @ A6 )
=> ( member @ nat @ ( gcd_gcd @ nat @ A5 @ B4 ) @ A6 ) ) )
=> ( ( A6
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( gcd_Gcd @ nat @ A6 ) @ A6 ) ) ) ).
% Gcd_in
thf(fact_5319_GreatestI__nat,axiom,
! [P: nat > $o,K2: nat,B2: nat] :
( ( P @ K2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_5320_Greatest__le__nat,axiom,
! [P: nat > $o,K2: nat,B2: nat] :
( ( P @ K2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ( ord_less_eq @ nat @ K2 @ ( order_Greatest @ nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_5321_GreatestI__ex__nat,axiom,
! [P: nat > $o,B2: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_5322_bezout__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [X5: nat,Y4: nat] :
( ( times_times @ nat @ A3 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% bezout_nat
thf(fact_5323_bezout__gcd__nat_H,axiom,
! [B2: nat,A3: nat] :
? [X5: nat,Y4: nat] :
( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( times_times @ nat @ A3 @ X5 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ A3 @ X5 ) @ ( times_times @ nat @ B2 @ Y4 ) )
= ( gcd_gcd @ nat @ A3 @ B2 ) ) )
| ( ( ord_less_eq @ nat @ ( times_times @ nat @ A3 @ Y4 ) @ ( times_times @ nat @ B2 @ X5 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X5 ) @ ( times_times @ nat @ A3 @ Y4 ) )
= ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_5324_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B2 @ A3 )
| ( ord_less_eq @ A @ D3 @ C3 )
| ( ord_less_eq @ A @ B2 @ C3 )
| ( ord_less_eq @ A @ D3 @ A3 ) ) ) ) ).
% Ioc_disjoint
thf(fact_5325_Sup__eq__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A )
= ( ^ [A7: set @ A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [B8: A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ord_less_eq @ A @ X4 @ B8 ) ) ) ) ) ) ) ).
% Sup_eq_Inf
thf(fact_5326_Inf__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A )
= ( ^ [A7: set @ A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [B8: A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ord_less_eq @ A @ B8 @ X4 ) ) ) ) ) ) ) ).
% Inf_eq_Sup
thf(fact_5327_ivl__disj__un__two_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_5328_open__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A,X3: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( member @ A @ X3 @ S3 )
=> ( ( ord_less @ A @ Y @ X3 )
=> ? [B4: A] :
( ( ord_less @ A @ B4 @ X3 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ B4 @ X3 ) @ S3 ) ) ) ) ) ) ).
% open_left
thf(fact_5329_ivl__disj__int__two_I8_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(8)
thf(fact_5330_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_5331_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_5332_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_5333_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_5334_ivl__disj__int__two_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(2)
thf(fact_5335_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A,Q: A > $o] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ A @ Y6 @ X5 ) )
=> ( Q @ X5 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_5336_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ( order_Greatest @ A @ P )
= X3 ) ) ) ) ).
% Greatest_equality
thf(fact_5337_sum_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( G3 @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum.head
thf(fact_5338_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G3 @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.head
thf(fact_5339_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_5340_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_5341_ivl__disj__un__two__touch_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_5342_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_5343_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs: list @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [I4: nat] :
( ( Uu3
= ( nth @ A @ Xs @ I4 ) )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_5344_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A8: A,B8: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A8 @ B8 ) @ ( insert2 @ A @ A8 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_5345_ivl__disj__un__two_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_5346_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_5347_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_5348_ivl__disj__un__two_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_5349_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_5350_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_5351_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg3 )
= ( ( Deg = Deg3 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_5352_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
=> ( ( ? [Uu: $o,Uv: $o] :
( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( Xa2
= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ 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 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_5353_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X3 @ Xa2 )
=> ( ( ? [Uu: $o,Uv: $o] :
( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( Xa2
!= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X @ ( 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 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_5354_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) )
=> ( Xa2
= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( 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 ) @ Xa2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ 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 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_5355_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) )
=> ( Xa2
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( 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 ) @ Xa2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X @ ( 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 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_5356_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Y
= ( Xa2
= ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( Y
= ( ( Deg2 = Xa2 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ 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 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_5357_finite__Inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A6: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A6 ) )
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu3
= ( image2 @ ( set @ A ) @ A @ F4 @ A6 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A6 )
=> ( member @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% finite_Inf_Sup
thf(fact_5358_Sup__Inf__le,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ ( set @ A )] :
( ord_less_eq @ A
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu3
= ( image2 @ ( set @ A ) @ A @ F4 @ A6 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A6 )
=> ( member @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ) )
@ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A6 ) ) ) ) ).
% Sup_Inf_le
thf(fact_5359_Inf__Sup__le,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A6: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A6 ) )
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu3
= ( image2 @ ( set @ A ) @ A @ F4 @ A6 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A6 )
=> ( member @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% Inf_Sup_le
thf(fact_5360_Pow__Compl,axiom,
! [A: $tType,A6: set @ A] :
( ( pow2 @ A @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [B6: set @ A] :
( ( Uu3
= ( uminus_uminus @ ( set @ A ) @ B6 ) )
& ( member @ ( set @ A ) @ A6 @ ( pow2 @ A @ B6 ) ) ) ) ) ).
% Pow_Compl
thf(fact_5361_Inf__filter__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( filter @ A ) )
= ( ^ [S6: set @ ( filter @ A )] :
( complete_Sup_Sup @ ( filter @ A )
@ ( collect @ ( filter @ A )
@ ^ [F9: filter @ A] :
! [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ S6 )
=> ( ord_less_eq @ ( filter @ A ) @ F9 @ X4 ) ) ) ) ) ) ).
% Inf_filter_def
thf(fact_5362_gcd__nat_Opelims,axiom,
! [X3: nat,Xa2: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) )
=> ~ ( ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y = X3 ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_5363_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S3: set @ A] :
( ! [A5: A,B4: A,X5: A] :
( ( member @ A @ A5 @ S3 )
=> ( ( member @ A @ B4 @ S3 )
=> ( ( ord_less_eq @ A @ A5 @ X5 )
=> ( ( ord_less_eq @ A @ X5 @ B4 )
=> ( member @ A @ X5 @ S3 ) ) ) ) )
=> ? [A5: A,B4: A] :
( ( S3
= ( bot_bot @ ( set @ A ) ) )
| ( S3
= ( top_top @ ( set @ A ) ) )
| ( S3
= ( set_ord_lessThan @ A @ B4 ) )
| ( S3
= ( set_ord_atMost @ A @ B4 ) )
| ( S3
= ( set_ord_greaterThan @ A @ A5 ) )
| ( S3
= ( set_ord_atLeast @ A @ A5 ) )
| ( S3
= ( set_or5935395276787703475ssThan @ A @ A5 @ B4 ) )
| ( S3
= ( set_or3652927894154168847AtMost @ A @ A5 @ B4 ) )
| ( S3
= ( set_or7035219750837199246ssThan @ A @ A5 @ B4 ) )
| ( S3
= ( set_or1337092689740270186AtMost @ A @ A5 @ B4 ) ) ) ) ) ).
% interval_cases
thf(fact_5364_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),N3: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N3 )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N3 )
& ? [Xys: list @ A,X4: A,Y3: A,Xs6: list @ A,Ys7: list @ A] :
( ( Xs
= ( append @ A @ Xys @ ( cons @ A @ X4 @ Xs6 ) ) )
& ( Ys3
= ( append @ A @ Xys @ ( cons @ A @ Y3 @ Ys7 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R5 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_5365_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K2: A] :
( ( member @ A @ I @ ( set_ord_atLeast @ A @ K2 ) )
= ( ord_less_eq @ A @ K2 @ I ) ) ) ).
% atLeast_iff
thf(fact_5366_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_5367_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X3 ) @ ( set_ord_atLeast @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% atLeast_subset_iff
thf(fact_5368_image__add__atLeast,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K2 ) @ ( set_ord_atLeast @ A @ I ) )
= ( set_ord_atLeast @ A @ ( plus_plus @ A @ K2 @ I ) ) ) ) ).
% image_add_atLeast
thf(fact_5369_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_5370_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less_eq @ A @ L2 ) ) ) ) ) ).
% atLeast_def
thf(fact_5371_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_5372_atLeast__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ( set_ord_atLeast @ A @ X3 )
= ( top_top @ ( set @ A ) ) )
= ( X3
= ( bot_bot @ A ) ) ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_5373_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_5374_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_5375_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_5376_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_5377_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_5378_atLeast__Suc__greaterThan,axiom,
! [K2: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K2 ) )
= ( set_ord_greaterThan @ nat @ K2 ) ) ).
% atLeast_Suc_greaterThan
thf(fact_5379_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_5380_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% Ici_subset_Ioi_iff
thf(fact_5381_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_5382_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_5383_lexn__length,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),N: nat] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( lexn @ A @ R2 @ N ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= N )
& ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ) ).
% lexn_length
thf(fact_5384_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_5385_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_5386_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_5387_atLeast__Suc,axiom,
! [K2: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K2 ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K2 ) @ ( insert2 @ nat @ K2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast_Suc
thf(fact_5388_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_5389_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_5390_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ? [Xys: list @ A,X4: A,Y3: A,Xs6: list @ A,Ys7: list @ A] :
( ( Xs
= ( append @ A @ Xys @ ( cons @ A @ X4 @ Xs6 ) ) )
& ( Ys3
= ( append @ A @ Xys @ ( cons @ A @ Y3 @ Ys7 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R5 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_5391_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G3: B > A,Y8: set @ B,X6: set @ A,F6: filter @ B,F3: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G3 @ Y8 ) @ X6 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( member @ B @ X4 @ Y8 )
@ F6 )
=> ( ( map_filter_on @ A @ C @ X6 @ F3 @ ( map_filter_on @ B @ A @ Y8 @ G3 @ F6 ) )
= ( map_filter_on @ B @ C @ Y8 @ ( comp @ A @ C @ B @ F3 @ G3 ) @ F6 ) ) ) ) ).
% map_filter_on_comp
thf(fact_5392_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( sup_sup @ A @ X4 ) @ Y3 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_5393_Sup__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A] :
( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% Sup_fin.singleton
thf(fact_5394_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_5395_Cons__in__lex,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y @ Ys ) ) @ ( lex @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys ) ) )
| ( ( X3 = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( lex @ A @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_5396_Sup__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ord_less_eq @ A @ A3 @ ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_5397_Nil2__notin__lex,axiom,
! [A: $tType,Xs2: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) @ ( lex @ A @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_5398_Nil__notin__lex,axiom,
! [A: $tType,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) @ ( lex @ A @ R2 ) ) ).
% Nil_notin_lex
thf(fact_5399_lex__append__leftI,axiom,
! [A: $tType,Ys: list @ A,Zs2: list @ A,R2: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys ) @ ( append @ A @ Xs2 @ Zs2 ) ) @ ( lex @ A @ R2 ) ) ) ).
% lex_append_leftI
thf(fact_5400_eventually__map__filter__on,axiom,
! [B: $tType,A: $tType,X6: set @ A,F6: filter @ A,P: B > $o,F3: A > B] :
( ( eventually @ A
@ ^ [X4: A] : ( member @ A @ X4 @ X6 )
@ F6 )
=> ( ( eventually @ B @ P @ ( map_filter_on @ A @ B @ X6 @ F3 @ F6 ) )
= ( eventually @ A
@ ^ [X4: A] :
( ( P @ ( F3 @ X4 ) )
& ( member @ A @ X4 @ X6 ) )
@ F6 ) ) ) ).
% eventually_map_filter_on
thf(fact_5401_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ X3 )
=> ! [A18: A] :
( ( member @ A @ A18 @ A6 )
=> ( ord_less_eq @ A @ A18 @ X3 ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_5402_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ( ord_less_eq @ A @ A5 @ X3 ) )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ X3 ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_5403_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ X3 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ X4 @ X3 ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_5404_Sup__fin__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A6 )
= ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% Sup_fin_Sup
thf(fact_5405_cSup__eq__Sup__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= ( lattic5882676163264333800up_fin @ A @ X6 ) ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_5406_Sup__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( lattic5882676163264333800up_fin @ A @ A6 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Sup_fin.infinite
thf(fact_5407_lex__append__leftD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys: list @ A,Zs2: list @ A] :
( ! [X5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys ) @ ( append @ A @ Xs2 @ Zs2 ) ) @ ( lex @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_5408_lex__append__left__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys: list @ A,Zs2: list @ A] :
( ! [X5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys ) @ ( append @ A @ Xs2 @ Zs2 ) ) @ ( lex @ A @ R2 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_5409_lex__append__rightI,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),Vs: list @ A,Us: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( lex @ A @ R2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Us ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Us ) @ ( append @ A @ Ys @ Vs ) ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_rightI
thf(fact_5410_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ B5 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_5411_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H: A > A,N5: set @ A] :
( ! [X5: A,Y4: A] :
( ( H @ ( sup_sup @ A @ X5 @ Y4 ) )
= ( sup_sup @ A @ ( H @ X5 ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite2 @ A @ N5 )
=> ( ( N5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic5882676163264333800up_fin @ A @ N5 ) )
= ( lattic5882676163264333800up_fin @ A @ ( image2 @ A @ A @ H @ N5 ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_5412_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B5 ) @ ( lattic5882676163264333800up_fin @ A @ A6 ) )
= ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_5413_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y4: A] : ( member @ A @ ( sup_sup @ A @ X5 @ Y4 ) @ ( insert2 @ A @ X5 @ ( insert2 @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ A6 ) ) ) ) ) ).
% Sup_fin.closed
thf(fact_5414_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X3 @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_5415_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ B5 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_5416_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X3 @ A6 ) ) ) ) ).
% Sup_fin.eq_fold
thf(fact_5417_inf__Sup1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A6 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A8: A] :
( ( Uu3
= ( inf_inf @ A @ X3 @ A8 ) )
& ( member @ A @ A8 @ A6 ) ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_5418_inf__Sup2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ B5 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A8: A,B8: A] :
( ( Uu3
= ( inf_inf @ A @ A8 @ B8 ) )
& ( member @ A @ A8 @ A6 )
& ( member @ A @ B8 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_5419_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X3 @ A6 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_5420_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A6 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A6 )
= ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_5421_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
@ ^ [Xs: list @ A,Ys3: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( lex @ A @ R5 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_5422_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( inf_inf @ A @ X4 ) @ Y3 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_5423_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq: A > A > $o,P4: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P4 @ X4 )
& ! [Y3: A] :
( ( P4 @ Y3 )
=> ( Less_eq @ X4 @ Y3 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_5424_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A] :
( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% Inf_fin.singleton
thf(fact_5425_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R2 ) )
= ( Ns
!= ( nil @ A ) ) ) ).
% Nil_lenlex_iff1
thf(fact_5426_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_5427_ord_OLeast_Ocong,axiom,
! [A: $tType] :
( ( least @ A )
= ( least @ A ) ) ).
% ord.Least.cong
thf(fact_5428_Inf__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ A3 ) ) ) ) ).
% Inf_fin.coboundedI
thf(fact_5429_lenlex__irreflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ! [X5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Xs2 ) @ ( lenlex @ A @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_5430_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R2 ) ) ).
% Nil_lenlex_iff2
thf(fact_5431_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ X3 @ X4 ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_5432_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ( ord_less_eq @ A @ X3 @ A5 ) )
=> ( ord_less_eq @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_5433_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
=> ! [A18: A] :
( ( member @ A @ A18 @ A6 )
=> ( ord_less_eq @ A @ X3 @ A18 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_5434_Inf__fin__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A6 )
= ( complete_Inf_Inf @ A @ A6 ) ) ) ) ) ).
% Inf_fin_Inf
thf(fact_5435_cInf__eq__Inf__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= ( lattic7752659483105999362nf_fin @ A @ X6 ) ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_5436_Inf__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( lattic7752659483105999362nf_fin @ A @ A6 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Inf_fin.infinite
thf(fact_5437_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B5 ) @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_5438_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_5439_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H: A > A,N5: set @ A] :
( ! [X5: A,Y4: A] :
( ( H @ ( inf_inf @ A @ X5 @ Y4 ) )
= ( inf_inf @ A @ ( H @ X5 ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite2 @ A @ N5 )
=> ( ( N5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic7752659483105999362nf_fin @ A @ N5 ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image2 @ A @ A @ H @ N5 ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_5440_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B5 ) @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
= ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_5441_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X3 @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_5442_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y4: A] : ( member @ A @ ( inf_inf @ A @ X5 @ Y4 ) @ ( insert2 @ A @ X5 @ ( insert2 @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ A6 ) ) ) ) ) ).
% Inf_fin.closed
thf(fact_5443_lenlex__append1,axiom,
! [A: $tType,Us: list @ A,Xs2: list @ A,R: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs2 ) @ ( lenlex @ A @ R ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs ) @ ( append @ A @ Xs2 @ Ys ) ) @ ( lenlex @ A @ R ) ) ) ) ).
% lenlex_append1
thf(fact_5444_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ ( lattic7752659483105999362nf_fin @ A @ B5 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_5445_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_5446_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X3 @ A6 ) ) ) ) ).
% Inf_fin.eq_fold
thf(fact_5447_sup__Inf2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ ( lattic7752659483105999362nf_fin @ A @ B5 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A8: A,B8: A] :
( ( Uu3
= ( sup_sup @ A @ A8 @ B8 ) )
& ( member @ A @ A8 @ A6 )
& ( member @ A @ B8 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_5448_sup__Inf1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A8: A] :
( ( Uu3
= ( sup_sup @ A @ X3 @ A8 ) )
& ( member @ A @ A8 @ A6 ) ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_5449_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A6 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A6 )
= ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_5450_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X3 @ A6 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_5451_Cons__lenlex__iff,axiom,
! [A: $tType,M2: A,Ms: list @ A,N: A,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M2 @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R2 ) )
= ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
| ( ( ( size_size @ ( list @ A ) @ Ms )
= ( size_size @ ( list @ A ) @ Ns ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M2 @ N ) @ R2 ) )
| ( ( M2 = N )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_5452_lexord__def,axiom,
! [A: $tType] :
( ( lexord @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [X4: list @ A,Y3: list @ A] :
? [A8: A,V5: list @ A] :
( ( Y3
= ( append @ A @ X4 @ ( cons @ A @ A8 @ V5 ) ) )
| ? [U2: list @ A,B8: A,C6: A,W3: list @ A,Z4: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ C6 ) @ R5 )
& ( X4
= ( append @ A @ U2 @ ( cons @ A @ B8 @ W3 ) ) )
& ( Y3
= ( append @ A @ U2 @ ( cons @ A @ C6 @ Z4 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_5453_eventually__filtercomap__at__topological,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ B )
=> ! [P: A > $o,F3: A > B,A6: B,B5: set @ B] :
( ( eventually @ A @ P @ ( filtercomap @ A @ B @ F3 @ ( topolo174197925503356063within @ B @ A6 @ B5 ) ) )
= ( ? [S6: set @ B] :
( ( topolo1002775350975398744n_open @ B @ S6 )
& ( member @ B @ A6 @ S6 )
& ! [X4: A] :
( ( member @ B @ ( F3 @ X4 ) @ ( minus_minus @ ( set @ B ) @ ( inf_inf @ ( set @ B ) @ S6 @ B5 ) @ ( insert2 @ B @ A6 @ ( bot_bot @ ( set @ B ) ) ) ) )
=> ( P @ X4 ) ) ) ) ) ) ).
% eventually_filtercomap_at_topological
thf(fact_5454_dual__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Min @ A
@ ^ [X4: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X4 ) )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% dual_Min
thf(fact_5455_filterlim__filtercomap,axiom,
! [A: $tType,B: $tType,F3: A > B,F6: filter @ B] : ( filterlim @ A @ B @ F3 @ F6 @ ( filtercomap @ A @ B @ F3 @ F6 ) ) ).
% filterlim_filtercomap
thf(fact_5456_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_5457_eventually__filtercomapI,axiom,
! [B: $tType,A: $tType,P: A > $o,F6: filter @ A,F3: B > A] :
( ( eventually @ A @ P @ F6 )
=> ( eventually @ B
@ ^ [X4: B] : ( P @ ( F3 @ X4 ) )
@ ( filtercomap @ B @ A @ F3 @ F6 ) ) ) ).
% eventually_filtercomapI
thf(fact_5458_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_5459_lexord__cons__cons,axiom,
! [A: $tType,A3: A,X3: list @ A,B2: A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A3 @ X3 ) @ ( cons @ A @ B2 @ Y ) ) @ ( lexord @ A @ R2 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
| ( ( A3 = B2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_5460_lexord__Nil__left,axiom,
! [A: $tType,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y ) @ ( lexord @ A @ R2 ) )
= ( ? [A8: A,X4: list @ A] :
( Y
= ( cons @ A @ A8 @ X4 ) ) ) ) ).
% lexord_Nil_left
thf(fact_5461_filtercomap__inf,axiom,
! [A: $tType,B: $tType,F3: A > B,F13: filter @ B,F24: filter @ B] :
( ( filtercomap @ A @ B @ F3 @ ( inf_inf @ ( filter @ B ) @ F13 @ F24 ) )
= ( inf_inf @ ( filter @ A ) @ ( filtercomap @ A @ B @ F3 @ F13 ) @ ( filtercomap @ A @ B @ F3 @ F24 ) ) ) ).
% filtercomap_inf
thf(fact_5462_filtercomap__ident,axiom,
! [A: $tType,F6: filter @ A] :
( ( filtercomap @ A @ A
@ ^ [X4: A] : X4
@ F6 )
= F6 ) ).
% filtercomap_ident
thf(fact_5463_filtercomap__filtercomap,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,G3: B > C,F6: filter @ C] :
( ( filtercomap @ A @ B @ F3 @ ( filtercomap @ B @ C @ G3 @ F6 ) )
= ( filtercomap @ A @ C
@ ^ [X4: A] : ( G3 @ ( F3 @ X4 ) )
@ F6 ) ) ).
% filtercomap_filtercomap
thf(fact_5464_filtercomap__mono,axiom,
! [B: $tType,A: $tType,F6: filter @ A,F11: filter @ A,F3: B > A] :
( ( ord_less_eq @ ( filter @ A ) @ F6 @ F11 )
=> ( ord_less_eq @ ( filter @ B ) @ ( filtercomap @ B @ A @ F3 @ F6 ) @ ( filtercomap @ B @ A @ F3 @ F11 ) ) ) ).
% filtercomap_mono
thf(fact_5465_eventually__filtercomap,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: A > B,F6: filter @ B] :
( ( eventually @ A @ P @ ( filtercomap @ A @ B @ F3 @ F6 ) )
= ( ? [Q6: B > $o] :
( ( eventually @ B @ Q6 @ F6 )
& ! [X4: A] :
( ( Q6 @ ( F3 @ X4 ) )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap
thf(fact_5466_filterlim__filtercomap__iff,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B,G3: B > C,G7: filter @ C,F6: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( filtercomap @ B @ C @ G3 @ G7 ) @ F6 )
= ( filterlim @ A @ C @ ( comp @ B @ C @ A @ G3 @ F3 ) @ G7 @ F6 ) ) ).
% filterlim_filtercomap_iff
thf(fact_5467_lexord__linear,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: list @ A,Y: list @ A] :
( ! [A5: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R2 )
| ( A5 = B4 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) )
| ( X3 = Y )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X3 ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_linear
thf(fact_5468_lexord__irreflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ! [X5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Xs2 ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_irreflexive
thf(fact_5469_filterlim__iff__le__filtercomap,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F4: A > B,F9: filter @ B,G9: filter @ A] : ( ord_less_eq @ ( filter @ A ) @ G9 @ ( filtercomap @ A @ B @ F4 @ F9 ) ) ) ) ).
% filterlim_iff_le_filtercomap
thf(fact_5470_filtercomap__neq__bot,axiom,
! [A: $tType,B: $tType,F6: filter @ A,F3: B > A] :
( ! [P7: A > $o] :
( ( eventually @ A @ P7 @ F6 )
=> ? [X: B] : ( P7 @ ( F3 @ X ) ) )
=> ( ( filtercomap @ B @ A @ F3 @ F6 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtercomap_neq_bot
thf(fact_5471_lexord__Nil__right,axiom,
! [A: $tType,X3: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ ( nil @ A ) ) @ ( lexord @ A @ R2 ) ) ).
% lexord_Nil_right
thf(fact_5472_lexord__append__leftI,axiom,
! [A: $tType,U: list @ A,V2: list @ A,R2: set @ ( product_prod @ A @ A ),X3: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X3 @ U ) @ ( append @ A @ X3 @ V2 ) ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_append_leftI
thf(fact_5473_filtercomap__sup,axiom,
! [A: $tType,B: $tType,F3: A > B,F13: filter @ B,F24: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( sup_sup @ ( filter @ A ) @ ( filtercomap @ A @ B @ F3 @ F13 ) @ ( filtercomap @ A @ B @ F3 @ F24 ) ) @ ( filtercomap @ A @ B @ F3 @ ( sup_sup @ ( filter @ B ) @ F13 @ F24 ) ) ) ).
% filtercomap_sup
thf(fact_5474_filtercomap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,F6: C > ( filter @ B ),B5: set @ C] :
( ( filtercomap @ A @ B @ F3 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F6 @ B5 ) ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B8: C] : ( filtercomap @ A @ B @ F3 @ ( F6 @ B8 ) )
@ B5 ) ) ) ).
% filtercomap_INF
thf(fact_5475_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 ) ) )
= ( ? [N6: A] :
! [X4: B] :
( ( ord_less_eq @ A @ N6 @ ( F3 @ X4 ) )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_top_linorder
thf(fact_5476_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 ) ) )
= ( ? [N6: A] :
! [X4: B] :
( ( ord_less @ A @ N6 @ ( F3 @ X4 ) )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_top_dense
thf(fact_5477_filtercomap__neq__bot__surj,axiom,
! [A: $tType,B: $tType,F6: filter @ A,F3: B > A] :
( ( F6
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( filtercomap @ B @ A @ F3 @ F6 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% filtercomap_neq_bot_surj
thf(fact_5478_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 ) ) )
= ( ? [N6: A] :
! [X4: B] :
( ( ord_less_eq @ A @ ( F3 @ X4 ) @ N6 )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_linorder
thf(fact_5479_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 ) ) )
= ( ? [N6: A] :
! [X4: B] :
( ( ord_less @ A @ ( F3 @ X4 ) @ N6 )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_dense
thf(fact_5480_lexord__partial__trans,axiom,
! [A: $tType,Xs2: list @ A,R2: set @ ( product_prod @ A @ A ),Ys: list @ A,Zs2: list @ A] :
( ! [X5: A,Y4: A,Z3: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Z3 ) @ R2 ) ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( lexord @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lexord @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Zs2 ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_5481_lexord__append__leftD,axiom,
! [A: $tType,X3: list @ A,U: list @ A,V2: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X3 @ U ) @ ( append @ A @ X3 @ V2 ) ) @ ( lexord @ A @ R2 ) )
=> ( ! [A5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_append_leftD
thf(fact_5482_lexord__append__rightI,axiom,
! [A: $tType,Y: list @ A,X3: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ? [B10: A,Z5: list @ A] :
( Y
= ( cons @ A @ B10 @ Z5 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ ( append @ A @ X3 @ Y ) ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_append_rightI
thf(fact_5483_lexord__sufE,axiom,
! [A: $tType,Xs2: list @ A,Zs2: list @ A,Ys: list @ A,Qs: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Zs2 ) @ ( append @ A @ Ys @ Qs ) ) @ ( lexord @ A @ R2 ) )
=> ( ( Xs2 != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs2 )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( lexord @ A @ R2 ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_5484_lexord__lex,axiom,
! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lex @ A @ R2 ) )
= ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) )
& ( ( size_size @ ( list @ A ) @ X3 )
= ( size_size @ ( list @ A ) @ Y ) ) ) ) ).
% lexord_lex
thf(fact_5485_filtercomap__SUP,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > C,F6: B > ( filter @ C ),B5: set @ B] :
( ord_less_eq @ ( filter @ A )
@ ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [B8: B] : ( filtercomap @ A @ C @ F3 @ ( F6 @ B8 ) )
@ B5 ) )
@ ( filtercomap @ A @ C @ F3 @ ( complete_Sup_Sup @ ( filter @ C ) @ ( image2 @ B @ ( filter @ C ) @ F6 @ B5 ) ) ) ) ).
% filtercomap_SUP
thf(fact_5486_lexord__append__left__rightI,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A ),U: list @ A,X3: list @ A,Y: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A3 @ X3 ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y ) ) ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_append_left_rightI
thf(fact_5487_lexord__same__pref__iff,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,Zs2: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys ) @ ( append @ A @ Xs2 @ Zs2 ) ) @ ( lexord @ A @ R2 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R2 ) )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_5488_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W: list @ A,R2: set @ ( product_prod @ A @ A ),V2: list @ A,Z2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W ) @ ( lexord @ A @ R2 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W ) @ ( size_size @ ( list @ A ) @ U ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V2 ) @ ( append @ A @ W @ Z2 ) ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_sufI
thf(fact_5489_List_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [R5: A > A > $o,Xs: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% List.lexordp_def
thf(fact_5490_upto_Opelims,axiom,
! [X3: int,Xa2: int,Y: list @ int] :
( ( ( upto @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X3 @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X3 @ Xa2 )
=> ( Y
= ( cons @ int @ X3 @ ( upto @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ int @ X3 @ Xa2 )
=> ( Y
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X3 @ Xa2 ) ) ) ) ) ).
% upto.pelims
thf(fact_5491_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_5492_length__upto,axiom,
! [I: int,J: int] :
( ( size_size @ ( list @ int ) @ ( upto @ I @ J ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I ) @ ( one_one @ int ) ) ) ) ).
% length_upto
thf(fact_5493_upto__rec__numeral_I1_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( numeral_numeral @ int @ M2 ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(1)
thf(fact_5494_upto__rec__numeral_I4_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(4)
thf(fact_5495_upto__rec__numeral_I3_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(3)
thf(fact_5496_upto__rec__numeral_I2_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( numeral_numeral @ int @ M2 ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(2)
thf(fact_5497_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,K5: Aa] :
( ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( summable @ Aa
@ ^ [N3: nat] : ( times_times @ Aa @ ( C3 @ N3 ) @ ( power_power @ Aa @ K5 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ Aa @ ( F3 @ A3 ) ) @ ( real_V7770717601297561774m_norm @ Aa @ K5 ) )
=> ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] :
( suminf @ Aa
@ ^ [N3: nat] : ( times_times @ Aa @ ( C3 @ N3 ) @ ( power_power @ Aa @ ( F3 @ X4 ) @ N3 ) ) ) ) ) ) ) ) ).
% isCont_powser'
thf(fact_5498_isCont__powser,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K5: A,X3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C3 @ N3 ) @ ( power_power @ A @ K5 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C3 @ N3 ) @ ( power_power @ A @ X4 @ N3 ) ) ) ) ) ) ) ).
% isCont_powser
thf(fact_5499_compactE__image,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,C4: set @ B,F3: B > ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ! [T5: B] :
( ( member @ B @ T5 @ C4 )
=> ( topolo1002775350975398744n_open @ A @ ( F3 @ T5 ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ C4 ) ) )
=> ~ ! [C8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C8 @ C4 )
=> ( ( finite_finite2 @ B @ C8 )
=> ~ ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ C8 ) ) ) ) ) ) ) ) ) ).
% compactE_image
thf(fact_5500_continuous__add,axiom,
! [B: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [F6: filter @ D,F3: D > B,G3: D > B] :
( ( topolo3448309680560233919inuous @ D @ B @ F6 @ F3 )
=> ( ( topolo3448309680560233919inuous @ D @ B @ F6 @ G3 )
=> ( topolo3448309680560233919inuous @ D @ B @ F6
@ ^ [X4: D] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ).
% continuous_add
thf(fact_5501_isCont__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A3: A,F3: A > B,G3: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G3 )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( product_Pair @ B @ C @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ).
% isCont_Pair
thf(fact_5502_continuous__power_H,axiom,
! [B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( topolo1898628316856586783d_mult @ B ) )
=> ! [F6: filter @ C,F3: C > B,G3: C > nat] :
( ( topolo3448309680560233919inuous @ C @ B @ F6 @ F3 )
=> ( ( topolo3448309680560233919inuous @ C @ nat @ F6 @ G3 )
=> ( topolo3448309680560233919inuous @ C @ B @ F6
@ ^ [X4: C] : ( power_power @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ).
% continuous_power'
thf(fact_5503_continuous__power,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F6: filter @ A,F3: A > B,N: nat] :
( ( topolo3448309680560233919inuous @ A @ B @ F6 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ F6
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N ) ) ) ) ).
% continuous_power
thf(fact_5504_continuous__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F6: filter @ A,F3: A > B,G3: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ F6 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ F6 @ G3 )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ F6
@ ^ [X4: A] : ( product_Pair @ B @ C @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ).
% continuous_Pair
thf(fact_5505_compact__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo2193935891317330818ompact @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% compact_empty
thf(fact_5506_IVT,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F3: A > B,A3: A,Y: B,B2: A] :
( ( ord_less_eq @ B @ ( F3 @ A3 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ! [X5: A] :
( ( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) @ F3 ) )
=> ? [X5: A] :
( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 )
& ( ( F3 @ X5 )
= Y ) ) ) ) ) ) ) ).
% IVT
thf(fact_5507_IVT2,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F3: A > B,B2: A,Y: B,A3: A] :
( ( ord_less_eq @ B @ ( F3 @ B2 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F3 @ A3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ! [X5: A] :
( ( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) @ F3 ) )
=> ? [X5: A] :
( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 )
& ( ( F3 @ X5 )
= Y ) ) ) ) ) ) ) ).
% IVT2
thf(fact_5508_compact__attains__sup,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ S3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ S3 )
=> ( ord_less_eq @ A @ Xa @ X5 ) ) ) ) ) ) ).
% compact_attains_sup
thf(fact_5509_compact__attains__inf,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ S3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ S3 )
=> ( ord_less_eq @ A @ X5 @ Xa ) ) ) ) ) ) ).
% compact_attains_inf
thf(fact_5510_isCont__add,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [A3: A,F3: A > B,G3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G3 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ).
% isCont_add
thf(fact_5511_isCont__power,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [A3: A,F3: A > B,N: nat] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N ) ) ) ) ).
% isCont_power
thf(fact_5512_isCont__eq__Lb,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B2: real,F3: real > A] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M8: A] :
( ! [X: real] :
( ( ( ord_less_eq @ real @ A3 @ X )
& ( ord_less_eq @ real @ X @ B2 ) )
=> ( ord_less_eq @ A @ M8 @ ( F3 @ X ) ) )
& ? [X5: real] :
( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 )
& ( ( F3 @ X5 )
= M8 ) ) ) ) ) ) ).
% isCont_eq_Lb
thf(fact_5513_isCont__eq__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B2: real,F3: real > A] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M8: A] :
( ! [X: real] :
( ( ( ord_less_eq @ real @ A3 @ X )
& ( ord_less_eq @ real @ X @ B2 ) )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ M8 ) )
& ? [X5: real] :
( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 )
& ( ( F3 @ X5 )
= M8 ) ) ) ) ) ) ).
% isCont_eq_Ub
thf(fact_5514_isCont__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B2: real,F3: real > A] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M8: A] :
! [X: real] :
( ( ( ord_less_eq @ real @ A3 @ X )
& ( ord_less_eq @ real @ X @ B2 ) )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ M8 ) ) ) ) ) ).
% isCont_bounded
thf(fact_5515_isCont__has__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B2: real,F3: real > A] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M8: A] :
( ! [X: real] :
( ( ( ord_less_eq @ real @ A3 @ X )
& ( ord_less_eq @ real @ X @ B2 ) )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ M8 ) )
& ! [N7: A] :
( ( ord_less @ A @ N7 @ M8 )
=> ? [X5: real] :
( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 )
& ( ord_less @ A @ N7 @ ( F3 @ X5 ) ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_5516_isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X3: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
= ( filterlim @ A @ B
@ ^ [H2: A] : ( F3 @ ( plus_plus @ A @ X3 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ X3 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% isCont_iff
thf(fact_5517_isCont__polynom,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: A,C3: nat > A,N: nat] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [W3: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ W3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% isCont_polynom
thf(fact_5518_isCont__powser__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,X3: A] :
( ! [Y4: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C3 @ N3 ) @ ( power_power @ A @ Y4 @ N3 ) ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C3 @ N3 ) @ ( power_power @ A @ X4 @ N3 ) ) ) ) ) ) ).
% isCont_powser_converges_everywhere
thf(fact_5519_isCont__If__ge,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,G3: A > B,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) @ G3 )
=> ( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( G3 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( if @ B @ ( ord_less_eq @ A @ X4 @ A3 ) @ ( G3 @ X4 ) @ ( F3 @ X4 ) ) ) ) ) ) ).
% isCont_If_ge
thf(fact_5520_compactE,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,T10: set @ ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ T10 ) )
=> ( ! [B7: set @ A] :
( ( member @ ( set @ A ) @ B7 @ T10 )
=> ( topolo1002775350975398744n_open @ A @ B7 ) )
=> ~ ! [T11: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ T11 @ T10 )
=> ( ( finite_finite2 @ ( set @ A ) @ T11 )
=> ~ ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ T11 ) ) ) ) ) ) ) ) ).
% compactE
thf(fact_5521_compactI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A] :
( ! [C7: set @ ( set @ A )] :
( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ C7 )
=> ( topolo1002775350975398744n_open @ A @ X ) )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ ( 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 ) @ S @ ( complete_Sup_Sup @ ( set @ A ) @ C9 ) ) ) ) )
=> ( topolo2193935891317330818ompact @ A @ S ) ) ) ).
% compactI
thf(fact_5522_compact__eq__Heine__Borel,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo2193935891317330818ompact @ A )
= ( ^ [S6: set @ A] :
! [C5: set @ ( set @ A )] :
( ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C5 )
=> ( topolo1002775350975398744n_open @ A @ X4 ) )
& ( ord_less_eq @ ( set @ A ) @ S6 @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) )
=> ? [D7: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ D7 @ C5 )
& ( finite_finite2 @ ( set @ A ) @ D7 )
& ( ord_less_eq @ ( set @ A ) @ S6 @ ( complete_Sup_Sup @ ( set @ A ) @ D7 ) ) ) ) ) ) ) ).
% compact_eq_Heine_Borel
thf(fact_5523_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs: list @ A,Ys3: list @ A] :
? [Us2: list @ A,Z4: A,Z9: A,Vs2: list @ A] :
( ( Xs
= ( append @ A @ Us2 @ ( cons @ A @ Z4 @ Vs2 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ Z9 ) @ R5 )
& ( Ys3
= ( append @ A @ Us2 @ ( cons @ A @ Z9 @ Vs2 ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_5524_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X4: B,Z4: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y3 ) )
@ Z4
@ B5 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_5525_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A7: set @ C,F4: C > A,G4: C > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A8: C] :
( ( Uu3
= ( product_Pair @ A @ B @ ( F4 @ A8 ) @ ( G4 @ A8 ) ) )
& ( member @ C @ A8 @ A7 ) ) ) ) ) ).
% image2_def
thf(fact_5526_Cons__listrel1__Cons,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
& ( Xs2 = Ys ) )
| ( ( X3 = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_5527_comp__fun__commute__filter__fold,axiom,
! [A: $tType,P: A > $o] :
( finite6289374366891150609ommute @ A @ ( set @ A )
@ ^ [X4: A,A16: set @ A] : ( if @ ( set @ A ) @ ( P @ X4 ) @ ( insert2 @ A @ X4 @ A16 ) @ A16 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_5528_listrel1I2,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),X3: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ X3 @ Ys ) ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1I2
thf(fact_5529_not__listrel1__Nil,axiom,
! [A: $tType,Xs2: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) @ ( listrel1 @ A @ R2 ) ) ).
% not_listrel1_Nil
thf(fact_5530_not__Nil__listrel1,axiom,
! [A: $tType,Xs2: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs2 ) @ ( listrel1 @ A @ R2 ) ) ).
% not_Nil_listrel1
thf(fact_5531_listrel1__eq__len,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_5532_append__listrel1I,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),Us: list @ A,Vs: list @ A] :
( ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel1 @ A @ R2 ) )
& ( Us = Vs ) )
| ( ( Xs2 = Ys )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Vs ) @ ( listrel1 @ A @ R2 ) ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Us ) @ ( append @ A @ Ys @ Vs ) ) @ ( listrel1 @ A @ R2 ) ) ) ).
% append_listrel1I
thf(fact_5533_Cons__listrel1E2,axiom,
! [A: $tType,Xs2: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R2 ) )
=> ( ! [X5: A] :
( ( Xs2
= ( cons @ A @ X5 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y ) @ R2 ) )
=> ~ ! [Zs: list @ A] :
( ( Xs2
= ( cons @ A @ Y @ Zs ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_5534_Cons__listrel1E1,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( ! [Y4: A] :
( ( Ys
= ( cons @ A @ Y4 @ Xs2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 ) )
=> ~ ! [Zs: list @ A] :
( ( Ys
= ( cons @ A @ X3 @ Zs ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Zs ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_5535_listrel1I1,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y @ Xs2 ) ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1I1
thf(fact_5536_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F3: B > A,X3: B,C3: C,G3: B > C,A6: set @ B] :
( ( B2
= ( F3 @ X3 ) )
=> ( ( C3
= ( G3 @ X3 ) )
=> ( ( member @ B @ X3 @ A6 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F3 @ G3 ) ) ) ) ) ).
% image2_eqI
thf(fact_5537_comp__fun__commute__relcomp__fold,axiom,
! [A: $tType,B: $tType,C: $tType,S3: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( finite6289374366891150609ommute @ ( product_prod @ C @ A ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ C @ A @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [X4: C,Y3: A,A7: set @ ( product_prod @ C @ B )] :
( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z4: B,A16: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y3 = W3 ) @ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X4 @ Z4 ) @ A16 ) @ A16 ) )
@ A7
@ S3 ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_5538_listrel1E,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ~ ! [X5: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ R2 )
=> ! [Us3: list @ A,Vs3: list @ A] :
( ( Xs2
= ( append @ A @ Us3 @ ( cons @ A @ X5 @ Vs3 ) ) )
=> ( Ys
!= ( append @ A @ Us3 @ ( cons @ A @ Y4 @ Vs3 ) ) ) ) ) ) ).
% listrel1E
thf(fact_5539_listrel1I,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs2: list @ A,Us: list @ A,Vs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( ( Xs2
= ( append @ A @ Us @ ( cons @ A @ X3 @ Vs ) ) )
=> ( ( Ys
= ( append @ A @ Us @ ( cons @ A @ Y @ Vs ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_5540_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs2: list @ A,X3: A,Ys: list @ A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel1 @ A @ R2 ) )
& ( X3 = Y ) )
| ( ( Xs2 = Ys )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_5541_listrel1__iff__update,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel1 @ A @ R2 ) )
= ( ? [Y3: A,N3: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs2 @ N3 ) @ Y3 ) @ R2 )
& ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( Ys
= ( list_update @ A @ Xs2 @ N3 @ Y3 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_5542_listrel1p__def,axiom,
! [A: $tType] :
( ( listrel1p @ A )
= ( ^ [R5: A > A > $o,Xs: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% listrel1p_def
thf(fact_5543_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A6: B > ( set @ A ),I: B,B5: set @ A,J5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( fun_upd @ B @ ( set @ A ) @ A6 @ I @ B5 ) @ J5 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ ( minus_minus @ ( set @ B ) @ J5 @ ( insert2 @ B @ I @ ( bot_bot @ ( set @ B ) ) ) ) ) ) @ ( if @ ( set @ A ) @ ( member @ B @ I @ J5 ) @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% UNION_fun_upd
thf(fact_5544_LIMSEQ__iff__nz,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X6 @ ( 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 )
& ! [N3: nat] :
( ( ord_less_eq @ nat @ No3 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ N3 ) @ L5 ) @ R5 ) ) ) ) ) ) ) ).
% LIMSEQ_iff_nz
thf(fact_5545_dist__add__cancel2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( real_V557655796197034286t_dist @ A @ ( plus_plus @ A @ B2 @ A3 ) @ ( plus_plus @ A @ C3 @ A3 ) )
= ( real_V557655796197034286t_dist @ A @ B2 @ C3 ) ) ) ).
% dist_add_cancel2
thf(fact_5546_dist__add__cancel,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( real_V557655796197034286t_dist @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ A3 @ C3 ) )
= ( real_V557655796197034286t_dist @ A @ B2 @ C3 ) ) ) ).
% dist_add_cancel
thf(fact_5547_comp__fun__commute__Image__fold,axiom,
! [B: $tType,A: $tType,S3: set @ A] :
( finite6289374366891150609ommute @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X4: A,Y3: B,A7: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X4 @ S3 ) @ ( insert2 @ B @ Y3 @ A7 ) @ A7 ) ) ) ).
% comp_fun_commute_Image_fold
thf(fact_5548_Cauchy__def,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M9: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M9 @ M5 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M9 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_def
thf(fact_5549_Cauchy__altdef2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [S7: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N6: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( S7 @ N3 ) @ ( S7 @ N6 ) ) @ E4 ) ) ) ) ) ) ).
% Cauchy_altdef2
thf(fact_5550_metric__CauchyD,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,E3: real] :
( ( topolo3814608138187158403Cauchy @ A @ X6 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M8: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M8 @ M3 )
=> ! [N9: nat] :
( ( ord_less_eq @ nat @ M8 @ N9 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M3 ) @ ( X6 @ N9 ) ) @ E3 ) ) ) ) ) ) ).
% metric_CauchyD
thf(fact_5551_metric__CauchyI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M10: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ M10 @ M )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M10 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M ) @ ( X6 @ N2 ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X6 ) ) ) ).
% metric_CauchyI
thf(fact_5552_dist__triangle__half__r,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y: A,X1: A,E3: real,X2: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X1 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X2 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X2 ) @ E3 ) ) ) ) ).
% dist_triangle_half_r
thf(fact_5553_dist__triangle__half__l,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X1: A,Y: A,E3: real,X2: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ Y ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X2 ) @ E3 ) ) ) ) ).
% dist_triangle_half_l
thf(fact_5554_dist__triangle__third,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X1: A,X2: A,E3: real,X32: A,X42: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X2 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ X32 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X32 @ X42 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X42 ) @ E3 ) ) ) ) ) ).
% dist_triangle_third
thf(fact_5555_Cauchy__altdef,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [F4: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M9: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M9 @ M5 )
=> ! [N3: nat] :
( ( ord_less @ nat @ M5 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F4 @ M5 ) @ ( F4 @ N3 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_altdef
thf(fact_5556_CauchyI_H,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M10: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ M10 @ M )
=> ! [N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M ) @ ( X6 @ N2 ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X6 ) ) ) ).
% CauchyI'
thf(fact_5557_fun__upd__image,axiom,
! [A: $tType,B: $tType,X3: B,A6: set @ B,F3: B > A,Y: A] :
( ( ( member @ B @ X3 @ A6 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F3 @ X3 @ Y ) @ A6 )
= ( insert2 @ A @ Y @ ( image2 @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert2 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ X3 @ A6 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F3 @ X3 @ Y ) @ A6 )
= ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ).
% fun_upd_image
thf(fact_5558_metric__LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,L5: A,R2: real] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ No @ N9 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ N9 ) @ L5 ) @ R2 ) ) ) ) ) ).
% metric_LIMSEQ_D
thf(fact_5559_metric__LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,L5: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No2: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No2 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ N2 ) @ L5 ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ).
% metric_LIMSEQ_I
thf(fact_5560_lim__sequentially,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No3 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ N3 ) @ L5 ) @ R5 ) ) ) ) ) ) ).
% lim_sequentially
thf(fact_5561_metric__Cauchy__iff2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [J3: nat] :
? [M9: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M9 @ M5 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M9 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_5562_comp__fun__commute__Pow__fold,axiom,
! [A: $tType] :
( finite6289374366891150609ommute @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A,A7: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A7 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X4 ) @ A7 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_5563_totally__bounded__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo6688025880775521714ounded @ A )
= ( ^ [S6: set @ A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [K3: set @ A] :
( ( finite_finite2 @ A @ K3 )
& ( ord_less_eq @ ( set @ A ) @ S6
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ A @ ( set @ A )
@ ^ [X4: A] :
( collect @ A
@ ^ [Y3: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ Y3 ) @ E4 ) )
@ K3 ) ) ) ) ) ) ) ) ).
% totally_bounded_metric
thf(fact_5564_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B ),X3: product_prod @ C @ A,X6: set @ ( product_prod @ C @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X3 @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S3 ) @ X6 )
= ( 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,Z4: B,A16: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X3 )
= W3 )
@ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X3 ) @ Z4 ) @ A16 )
@ A16 ) )
@ X6
@ S3 ) ) ) ).
% insert_relcomp_union_fold
thf(fact_5565_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B ),X3: product_prod @ C @ A,R: set @ ( product_prod @ C @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X3 @ R ) @ S3 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z4: B,A16: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X3 )
= W3 )
@ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X3 ) @ Z4 ) @ A16 )
@ A16 ) )
@ ( relcomp @ C @ A @ B @ R @ S3 )
@ S3 ) ) ) ).
% insert_relcomp_fold
thf(fact_5566_empty__upd__none,axiom,
! [B: $tType,A: $tType,X3: A] :
( ( fun_upd @ A @ ( option @ B )
@ ^ [X4: A] : ( none @ B )
@ X3
@ ( none @ B ) )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% empty_upd_none
thf(fact_5567_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_5568_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_5569_relcomp__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,S3: 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 ) ) @ S3 @ T4 ) @ R )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( relcomp @ A @ C @ B @ S3 @ R ) @ ( relcomp @ A @ C @ B @ T4 @ R ) ) ) ).
% relcomp_distrib2
thf(fact_5570_relcomp__distrib,axiom,
! [A: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ A @ C ),S3: set @ ( product_prod @ C @ B ),T4: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ R @ ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ S3 @ T4 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( relcomp @ A @ C @ B @ R @ S3 ) @ ( relcomp @ A @ C @ B @ R @ T4 ) ) ) ).
% relcomp_distrib
thf(fact_5571_image__map__upd,axiom,
! [B: $tType,A: $tType,X3: A,A6: set @ A,M2: A > ( option @ B ),Y: B] :
( ~ ( member @ A @ X3 @ A6 )
=> ( ( image2 @ A @ ( option @ B ) @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X3 @ ( some @ B @ Y ) ) @ A6 )
= ( image2 @ A @ ( option @ B ) @ M2 @ A6 ) ) ) ).
% image_map_upd
thf(fact_5572_totally__bounded__empty,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( topolo6688025880775521714ounded @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% totally_bounded_empty
thf(fact_5573_map__option__o__map__upd,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > B,M2: A > ( option @ C ),A3: A,B2: C] :
( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F3 ) @ ( fun_upd @ A @ ( option @ C ) @ M2 @ A3 @ ( some @ C @ B2 ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F3 ) @ M2 ) @ A3 @ ( some @ B @ ( F3 @ B2 ) ) ) ) ).
% map_option_o_map_upd
thf(fact_5574_map__upd__eqD1,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),A3: A,X3: B,N: A > ( option @ B ),Y: B] :
( ( ( fun_upd @ A @ ( option @ B ) @ M2 @ A3 @ ( some @ B @ X3 ) )
= ( fun_upd @ A @ ( option @ B ) @ N @ A3 @ ( some @ B @ Y ) ) )
=> ( X3 = Y ) ) ).
% map_upd_eqD1
thf(fact_5575_map__upd__triv,axiom,
! [A: $tType,B: $tType,T2: B > ( option @ A ),K2: B,X3: A] :
( ( ( T2 @ K2 )
= ( some @ A @ X3 ) )
=> ( ( fun_upd @ B @ ( option @ A ) @ T2 @ K2 @ ( some @ A @ X3 ) )
= T2 ) ) ).
% map_upd_triv
thf(fact_5576_map__upd__Some__unfold,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),A3: B,B2: A,X3: B,Y: A] :
( ( ( fun_upd @ B @ ( option @ A ) @ M2 @ A3 @ ( some @ A @ B2 ) @ X3 )
= ( some @ A @ Y ) )
= ( ( ( X3 = A3 )
& ( B2 = Y ) )
| ( ( X3 != A3 )
& ( ( M2 @ X3 )
= ( some @ A @ Y ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_5577_map__upd__nonempty,axiom,
! [B: $tType,A: $tType,T2: A > ( option @ B ),K2: A,X3: B] :
( ( fun_upd @ A @ ( option @ B ) @ T2 @ K2 @ ( some @ B @ X3 ) )
!= ( ^ [X4: A] : ( none @ B ) ) ) ).
% map_upd_nonempty
thf(fact_5578_relcomp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R4: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),S8: set @ ( product_prod @ B @ C ),S: set @ ( product_prod @ B @ C )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ R2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S8 @ S )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R4 @ S8 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_5579_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A3: A,C3: B,R2: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ C3 ) @ ( relcomp @ A @ C @ B @ R2 @ S ) )
=> ~ ! [B4: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ B4 ) @ R2 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B4 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_5580_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R2: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R2 @ S ) )
=> ~ ! [X5: A,Y4: C,Z3: B] :
( ( Xz
= ( product_Pair @ A @ B @ X5 @ Z3 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X5 @ Y4 ) @ R2 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y4 @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_5581_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R2: set @ ( product_prod @ A @ B ),C3: C,S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ C3 ) @ S )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ C3 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_5582_relcomp_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,A1: A,A22: C,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) )
= ( ? [A8: A,B8: B,C6: C] :
( ( A1 = A8 )
& ( A22 = C6 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B8 ) @ R2 )
& ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B8 @ C6 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_5583_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) )
=> ~ ! [B4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A1 @ B4 ) @ R2 )
=> ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B4 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_5584_O__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ A @ D ),S3: set @ ( product_prod @ D @ C ),T4: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( relcomp @ A @ D @ C @ R @ S3 ) @ T4 )
= ( relcomp @ A @ D @ B @ R @ ( relcomp @ D @ C @ B @ S3 @ T4 ) ) ) ).
% O_assoc
thf(fact_5585_totally__bounded__subset,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [S3: set @ A,T4: set @ A] :
( ( topolo6688025880775521714ounded @ A @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T4 @ S3 )
=> ( topolo6688025880775521714ounded @ A @ T4 ) ) ) ) ).
% totally_bounded_subset
thf(fact_5586_relcomp__UNION__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R2: D > ( set @ ( product_prod @ A @ C ) ),I5: set @ D,S: 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 ) ) @ S )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I4: D] : ( relcomp @ A @ C @ B @ ( R2 @ I4 ) @ S )
@ I5 ) ) ) ).
% relcomp_UNION_distrib2
thf(fact_5587_relcomp__UNION__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S: set @ ( product_prod @ A @ C ),R2: D > ( set @ ( product_prod @ C @ B ) ),I5: set @ D] :
( ( relcomp @ A @ C @ B @ S @ ( 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 @ S @ ( R2 @ I4 ) )
@ I5 ) ) ) ).
% relcomp_UNION_distrib
thf(fact_5588_relcomp__unfold,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcomp @ A @ C @ B )
= ( ^ [R5: set @ ( product_prod @ A @ C ),S7: set @ ( product_prod @ C @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Z4: B] :
? [Y3: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Y3 ) @ R5 )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y3 @ Z4 ) @ S7 ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_5589_finite__range__updI,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),A3: B,B2: A] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ ( fun_upd @ B @ ( option @ A ) @ F3 @ A3 @ ( some @ A @ B2 ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_updI
thf(fact_5590_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S3 )
=> ( ( relcomp @ A @ B @ C @ R @ S3 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [X4: A,Y3: B,A7: set @ ( product_prod @ A @ C )] :
( finite_fold @ ( product_prod @ B @ C ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ B @ C @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [W3: B,Z4: C,A16: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y3 = W3 ) @ ( insert2 @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Z4 ) @ A16 ) @ A16 ) )
@ A7
@ S3 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R ) ) ) ) ).
% relcomp_fold
thf(fact_5591_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B,M2: A > ( option @ B ),X3: A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M2 @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ Xs2 @ Ys ) @ X3 @ ( some @ B @ ( nth @ B @ Ys @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_5592_min__ext__compat,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S3 ) @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( min_ext @ A @ S3 ) @ ( 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_5593_relpow__finite__bounded1,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),K2: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K2 @ R )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R )
@ ( collect @ nat
@ ^ [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ) ) ).
% relpow_finite_bounded1
thf(fact_5594_map__upds__apply__nontin,axiom,
! [B: $tType,A: $tType,X3: A,Xs2: list @ A,F3: A > ( option @ B ),Ys: list @ B] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( map_upds @ A @ B @ F3 @ Xs2 @ Ys @ X3 )
= ( F3 @ X3 ) ) ) ).
% map_upds_apply_nontin
thf(fact_5595_fun__upds__append2__drop,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,M2: A > ( option @ B ),Zs2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M2 @ Xs2 @ ( append @ B @ Ys @ Zs2 ) )
= ( map_upds @ A @ B @ M2 @ Xs2 @ Ys ) ) ) ).
% fun_upds_append2_drop
thf(fact_5596_fun__upds__append__drop,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,M2: A > ( option @ B ),Zs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M2 @ ( append @ A @ Xs2 @ Zs2 ) @ Ys )
= ( map_upds @ A @ B @ M2 @ Xs2 @ Ys ) ) ) ).
% fun_upds_append_drop
thf(fact_5597_map__upds__list__update2__drop,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,I: nat,M2: A > ( option @ B ),Ys: list @ B,Y: B] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ I )
=> ( ( map_upds @ A @ B @ M2 @ Xs2 @ ( list_update @ B @ Ys @ I @ Y ) )
= ( map_upds @ A @ B @ M2 @ Xs2 @ Ys ) ) ) ).
% map_upds_list_update2_drop
thf(fact_5598_map__upds__Cons,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),A3: A,As2: list @ A,B2: B,Bs: list @ B] :
( ( map_upds @ A @ B @ M2 @ ( cons @ A @ A3 @ As2 ) @ ( cons @ B @ B2 @ Bs ) )
= ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ A3 @ ( some @ B @ B2 ) ) @ As2 @ Bs ) ) ).
% map_upds_Cons
thf(fact_5599_map__upds__twist,axiom,
! [A: $tType,B: $tType,A3: A,As2: list @ A,M2: A > ( option @ B ),B2: B,Bs: list @ B] :
( ~ ( member @ A @ A3 @ ( set2 @ A @ As2 ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ A3 @ ( some @ B @ B2 ) ) @ As2 @ Bs )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ As2 @ Bs ) @ A3 @ ( some @ B @ B2 ) ) ) ) ).
% map_upds_twist
thf(fact_5600_relpow__Suc__D2_H,axiom,
! [A: $tType,N: nat,R: set @ ( product_prod @ A @ A ),X: A,Y6: A,Z5: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y6 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z5 ) @ R ) )
=> ? [W2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ W2 ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W2 @ Z5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_5601_relpow__Suc__E,axiom,
! [A: $tType,X3: A,Z2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ R ) ) ) ).
% relpow_Suc_E
thf(fact_5602_relpow__Suc__I,axiom,
! [A: $tType,X3: A,Y: A,N: nat,R: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) ) ) ) ).
% relpow_Suc_I
thf(fact_5603_relpow__Suc__D2,axiom,
! [A: $tType,X3: A,Z2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) )
=> ? [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpow_Suc_D2
thf(fact_5604_relpow__Suc__E2,axiom,
! [A: $tType,X3: A,Z2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpow_Suc_E2
thf(fact_5605_relpow__Suc__I2,axiom,
! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A ),Z2: A,N: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) ) ) ) ).
% relpow_Suc_I2
thf(fact_5606_relpow__0__E,axiom,
! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R ) )
=> ( X3 = Y ) ) ).
% relpow_0_E
thf(fact_5607_relpow__0__I,axiom,
! [A: $tType,X3: A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R ) ) ).
% relpow_0_I
thf(fact_5608_relpow_Osimps_I2_J,axiom,
! [A: $tType,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R )
= ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) @ R ) ) ).
% relpow.simps(2)
thf(fact_5609_relpow__add,axiom,
! [A: $tType,M2: nat,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( plus_plus @ nat @ M2 @ N ) @ R )
= ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M2 @ R ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ).
% relpow_add
thf(fact_5610_relpowp__relpow__eq,axiom,
! [A: $tType,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( compow @ ( A > A > $o ) @ N
@ ^ [X4: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R ) )
= ( ^ [X4: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpowp_relpow_eq
thf(fact_5611_relpow__E2,axiom,
! [A: $tType,X3: A,Z2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X3 != Z2 ) )
=> ~ ! [Y4: A,M: nat] :
( ( N
= ( suc @ M ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M @ R ) ) ) ) ) ) ).
% relpow_E2
thf(fact_5612_relpow__E,axiom,
! [A: $tType,X3: A,Z2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X3 != Z2 ) )
=> ~ ! [Y4: A,M: nat] :
( ( N
= ( suc @ M ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ R ) ) ) ) ) ).
% relpow_E
thf(fact_5613_relpow__fun__conv,axiom,
! [A: $tType,A3: A,B2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
= ( ? [F4: nat > A] :
( ( ( F4 @ ( zero_zero @ nat ) )
= A3 )
& ( ( F4 @ N )
= B2 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_5614_relpow__finite__bounded,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),K2: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K2 @ R )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R )
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ).
% relpow_finite_bounded
thf(fact_5615_max__ext__compat,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S3 ) @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( max_ext @ A @ S3 ) @ ( 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_5616_ntrancl__def,axiom,
! [A: $tType] :
( ( transitive_ntrancl @ A )
= ( ^ [N3: 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 @ N3 ) ) ) ) ) ) ) ) ).
% ntrancl_def
thf(fact_5617_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 ) )
@ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R )
@ ( collect @ nat
@ ^ [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ) ).
% trancl_finite_eq_relpow
thf(fact_5618_trancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ R2 )
=> ~ ! [B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B4 ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A22 ) @ R2 ) ) ) ) ).
% trancl.cases
thf(fact_5619_trancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R2 ) )
= ( ? [A8: A,B8: A] :
( ( A1 = A8 )
& ( A22 = B8 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B8 ) @ R2 ) )
| ? [A8: A,B8: A,C6: A] :
( ( A1 = A8 )
& ( A22 = C6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B8 ) @ ( transitive_trancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ C6 ) @ R2 ) ) ) ) ).
% trancl.simps
thf(fact_5620_trancl_Or__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ).
% trancl.r_into_trancl
thf(fact_5621_tranclE,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ~ ! [C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ C2 @ B2 ) @ R2 ) ) ) ) ).
% tranclE
thf(fact_5622_trancl__trans,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_trans
thf(fact_5623_trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ R2 )
=> ( P @ Y4 ) )
=> ( ! [Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 )
=> ( ( P @ Y4 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% trancl_induct
thf(fact_5624_r__r__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% r_r_into_trancl
thf(fact_5625_converse__tranclE,axiom,
! [A: $tType,X3: A,Z2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ R2 )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ) ).
% converse_tranclE
thf(fact_5626_irrefl__trancl__rD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ! [X5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( X3 != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_5627_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_5628_trancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_5629_trancl__trans__induct,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),P: A > A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ R2 )
=> ( P @ X5 @ Y4 ) )
=> ( ! [X5: A,Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P @ X5 @ Y4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P @ Y4 @ Z3 )
=> ( P @ X5 @ Z3 ) ) ) ) )
=> ( P @ X3 @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_5630_converse__trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ B2 ) @ R2 )
=> ( P @ Y4 ) )
=> ( ! [Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P @ Z3 )
=> ( P @ Y4 ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% converse_trancl_induct
thf(fact_5631_trancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ! [A5: A,B4: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B4 ) ) @ R2 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B4 ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 )
=> ( ( P @ A5 @ B4 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_5632_max__ext__additive,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,R: set @ ( product_prod @ A @ A ),C4: set @ A,D4: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A6 @ B5 ) @ ( max_ext @ A @ R ) )
=> ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ C4 @ D4 ) @ ( max_ext @ A @ R ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ C4 ) @ ( sup_sup @ ( set @ A ) @ B5 @ D4 ) ) @ ( max_ext @ A @ R ) ) ) ) ).
% max_ext_additive
thf(fact_5633_less__eq,axiom,
! [M2: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M2 @ N ) @ ( transitive_trancl @ nat @ pred_nat ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% less_eq
thf(fact_5634_trancl__insert2,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y3: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ A3 ) @ ( transitive_trancl @ A @ R2 ) )
| ( X4 = A3 ) )
& ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y3 ) @ ( transitive_trancl @ A @ R2 ) )
| ( Y3 = B2 ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_5635_max__ext_Ocases,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R ) )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X: A] :
( ( member @ A @ X @ A1 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Xa3 ) @ R ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_5636_max__ext_Osimps,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R ) )
= ( ( finite_finite2 @ A @ A1 )
& ( finite_finite2 @ A @ A22 )
& ( A22
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A1 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R ) ) ) ) ) ).
% max_ext.simps
thf(fact_5637_max__ext_Omax__extI,axiom,
! [A: $tType,X6: set @ A,Y8: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X6 )
=> ( ( finite_finite2 @ A @ Y8 )
=> ( ( Y8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Y8 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa ) @ R ) ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X6 @ Y8 ) @ ( max_ext @ A @ R ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_5638_graph__map__upd,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),K2: A,V2: B] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K2 @ ( some @ B @ V2 ) ) )
= ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V2 ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K2 @ ( none @ B ) ) ) ) ) ).
% graph_map_upd
thf(fact_5639_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),X3: A,Y: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X3 @ ( some @ B @ Y ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M2 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_5640_restrict__map__upds,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,D4: set @ A,M2: A > ( option @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ D4 )
=> ( ( restrict_map @ A @ B @ ( map_upds @ A @ B @ M2 @ Xs2 @ Ys ) @ D4 )
= ( map_upds @ A @ B @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D4 @ ( set2 @ A @ Xs2 ) ) ) @ Xs2 @ Ys ) ) ) ) ).
% restrict_map_upds
thf(fact_5641_restrict__out,axiom,
! [A: $tType,B: $tType,X3: A,A6: set @ A,M2: A > ( option @ B )] :
( ~ ( member @ A @ X3 @ A6 )
=> ( ( restrict_map @ A @ B @ M2 @ A6 @ X3 )
= ( none @ B ) ) ) ).
% restrict_out
thf(fact_5642_restrict__map__empty,axiom,
! [B: $tType,A: $tType,D4: set @ A] :
( ( restrict_map @ A @ B
@ ^ [X4: A] : ( none @ B )
@ D4 )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% restrict_map_empty
thf(fact_5643_restrict__map__to__empty,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M2 @ ( bot_bot @ ( set @ A ) ) )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% restrict_map_to_empty
thf(fact_5644_graph__empty,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B
@ ^ [X4: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% graph_empty
thf(fact_5645_restrict__fun__upd,axiom,
! [B: $tType,A: $tType,X3: A,D4: set @ A,M2: A > ( option @ B ),Y: option @ B] :
( ( ( member @ A @ X3 @ D4 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X3 @ Y ) @ D4 )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X3 @ Y ) ) )
& ( ~ ( member @ A @ X3 @ D4 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X3 @ Y ) @ D4 )
= ( restrict_map @ A @ B @ M2 @ D4 ) ) ) ) ).
% restrict_fun_upd
thf(fact_5646_fun__upd__restrict__conv,axiom,
! [A: $tType,B: $tType,X3: A,D4: set @ A,M2: A > ( option @ B ),Y: option @ B] :
( ( member @ A @ X3 @ D4 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D4 ) @ X3 @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X3 @ Y ) ) ) ).
% fun_upd_restrict_conv
thf(fact_5647_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,X3: A,D4: set @ A,M2: A > ( option @ B )] :
( ( ( member @ A @ X3 @ D4 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D4 ) @ X3 @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
& ( ~ ( member @ A @ X3 @ D4 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D4 ) @ X3 @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ D4 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_5648_restrict__map__def,axiom,
! [B: $tType,A: $tType] :
( ( restrict_map @ A @ B )
= ( ^ [M5: A > ( option @ B ),A7: set @ A,X4: A] : ( if @ ( option @ B ) @ ( member @ A @ X4 @ A7 ) @ ( M5 @ X4 ) @ ( none @ B ) ) ) ) ).
% restrict_map_def
thf(fact_5649_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K2: A,V2: B,M2: A > ( option @ B ),A6: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V2 ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M2 @ A6 ) ) )
=> ( ( M2 @ K2 )
= ( some @ B @ V2 ) ) ) ).
% graph_restrictD(2)
thf(fact_5650_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K2: A,V2: B,M2: A > ( option @ B ),A6: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V2 ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M2 @ A6 ) ) )
=> ( member @ A @ K2 @ A6 ) ) ).
% graph_restrictD(1)
thf(fact_5651_in__graphD,axiom,
! [A: $tType,B: $tType,K2: A,V2: B,M2: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V2 ) @ ( graph @ A @ B @ M2 ) )
=> ( ( M2 @ K2 )
= ( some @ B @ V2 ) ) ) ).
% in_graphD
thf(fact_5652_in__graphI,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),K2: B,V2: A] :
( ( ( M2 @ K2 )
= ( some @ A @ V2 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K2 @ V2 ) @ ( graph @ B @ A @ M2 ) ) ) ).
% in_graphI
thf(fact_5653_restrict__map__insert,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B ),A3: A,A6: set @ A] :
( ( restrict_map @ A @ B @ F3 @ ( insert2 @ A @ A3 @ A6 ) )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ F3 @ A6 ) @ A3 @ ( F3 @ A3 ) ) ) ).
% restrict_map_insert
thf(fact_5654_graph__def,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M5: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A8: A,B8: B] :
( ( Uu3
= ( product_Pair @ A @ B @ A8 @ B8 ) )
& ( ( M5 @ A8 )
= ( some @ B @ B8 ) ) ) ) ) ) ).
% graph_def
thf(fact_5655_graph__fun__upd__None,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),K2: A] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K2 @ ( none @ B ) ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [E4: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ E4 @ ( graph @ A @ B @ M2 ) )
& ( ( product_fst @ A @ B @ E4 )
!= K2 ) ) ) ) ).
% graph_fun_upd_None
thf(fact_5656_fun__upd__restrict,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),D4: set @ A,X3: A,Y: option @ B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D4 ) @ X3 @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X3 @ Y ) ) ).
% fun_upd_restrict
thf(fact_5657_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B ),X3: A] :
( ( restrict_map @ A @ B @ F3 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( fun_upd @ A @ ( option @ B ) @ F3 @ X3 @ ( none @ B ) ) ) ).
% restrict_complement_singleton_eq
thf(fact_5658_max__extp__eq,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R5: A > A > $o,X4: set @ A,Y3: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X4 @ Y3 ) @ ( max_ext @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% max_extp_eq
thf(fact_5659_max__extp__max__ext__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( max_extp @ A
@ ^ [X4: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R ) )
= ( ^ [X4: set @ A,Y3: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X4 @ Y3 ) @ ( max_ext @ A @ R ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_5660_option_Orec__o__map,axiom,
! [B: $tType,C: $tType,A: $tType,G3: C,Ga: B > C,F3: A > B] :
( ( comp @ ( option @ B ) @ C @ ( option @ A ) @ ( rec_option @ C @ B @ G3 @ Ga ) @ ( map_option @ A @ B @ F3 ) )
= ( rec_option @ C @ A @ G3
@ ^ [X4: A] : ( Ga @ ( F3 @ X4 ) ) ) ) ).
% option.rec_o_map
thf(fact_5661_option_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F22: A > C,X2: A] :
( ( rec_option @ C @ A @ F1 @ F22 @ ( some @ A @ X2 ) )
= ( F22 @ X2 ) ) ).
% option.simps(7)
thf(fact_5662_option_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F22: A > C] :
( ( rec_option @ C @ A @ F1 @ F22 @ ( none @ A ) )
= F1 ) ).
% option.simps(6)
thf(fact_5663_max__ext__def,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ( max_extp @ A
@ ^ [X4: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R6 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_5664_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,A6: set @ A,B5: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ S3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A6 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( finite_fold @ A @ B @ F3 @ ( finite_fold @ A @ B @ F3 @ Z2 @ A6 ) @ B5 ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
thf(fact_5665_comp__fun__commute__on_Ofold__rec,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,A6: set @ A,X3: A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ S3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ A6 )
= ( F3 @ X3 @ ( finite_fold @ A @ B @ F3 @ Z2 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_rec
thf(fact_5666_comp__fun__commute__on_Ofold__insert__remove,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X3: A,A6: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X3 @ A6 ) @ S3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( insert2 @ A @ X3 @ A6 ) )
= ( F3 @ X3 @ ( finite_fold @ A @ B @ F3 @ Z2 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert_remove
thf(fact_5667_Finite__Set_Ofold__cong,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,G3: A > B > B,A6: set @ A,S: B,T2: B,B5: set @ A] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( finite4664212375090638736ute_on @ A @ B @ S3 @ G3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ S3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ( F3 @ X5 )
= ( G3 @ X5 ) ) )
=> ( ( S = T2 )
=> ( ( A6 = B5 )
=> ( ( finite_fold @ A @ B @ F3 @ S @ A6 )
= ( finite_fold @ A @ B @ G3 @ T2 @ B5 ) ) ) ) ) ) ) ) ) ).
% Finite_Set.fold_cong
thf(fact_5668_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
! [B: $tType,A: $tType,C: $tType,S3: set @ A,F3: A > B > B,G3: C > A,R: set @ C] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ G3 @ ( top_top @ ( set @ C ) ) ) @ S3 )
=> ( finite4664212375090638736ute_on @ C @ B @ R @ ( comp @ A @ ( B > B ) @ C @ F3 @ G3 ) ) ) ) ).
% comp_fun_commute_on.comp_comp_fun_commute_on
thf(fact_5669_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X3: A,A6: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X3 @ A6 ) @ S3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( F3 @ X3 @ ( finite_fold @ A @ B @ F3 @ Z2 @ A6 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X3 @ Z2 ) @ A6 ) ) ) ) ) ).
% comp_fun_commute_on.fold_fun_left_comm
thf(fact_5670_comp__fun__commute__on_Ofold__insert2,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X3: A,A6: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X3 @ A6 ) @ S3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X3 @ A6 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( insert2 @ A @ X3 @ A6 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X3 @ Z2 ) @ A6 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert2
thf(fact_5671_comp__fun__commute__on_Ofold__insert,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X3: A,A6: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X3 @ A6 ) @ S3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X3 @ A6 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( insert2 @ A @ X3 @ A6 ) )
= ( F3 @ X3 @ ( finite_fold @ A @ B @ F3 @ Z2 @ A6 ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert
thf(fact_5672_compact__imp__fip__image,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,I5: set @ B,F3: B > ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( topolo7761053866217962861closed @ A @ ( F3 @ I3 ) ) )
=> ( ! [I8: set @ B] :
( ( finite_finite2 @ B @ I8 )
=> ( ( ord_less_eq @ ( set @ B ) @ I8 @ I5 )
=> ( ( inf_inf @ ( set @ A ) @ S @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ I8 ) ) )
!= ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ S @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ I5 ) ) )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% compact_imp_fip_image
thf(fact_5673_Bseq__iff2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [X4: A] :
! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X6 @ N3 ) @ ( uminus_uminus @ A @ X4 ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff2
thf(fact_5674_Bseq__iff3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [N6: nat] :
! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X6 @ N3 ) @ ( uminus_uminus @ A @ ( X6 @ N6 ) ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff3
thf(fact_5675_closed__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo7761053866217962861closed @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% closed_empty
thf(fact_5676_closed__singleton,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [A3: A] : ( topolo7761053866217962861closed @ A @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% closed_singleton
thf(fact_5677_closed__insert,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [S3: set @ A,A3: A] :
( ( topolo7761053866217962861closed @ A @ S3 )
=> ( topolo7761053866217962861closed @ A @ ( insert2 @ A @ A3 @ S3 ) ) ) ) ).
% closed_insert
thf(fact_5678_closed__diagonal,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ( topolo7761053866217962861closed @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Y3: product_prod @ A @ A] :
? [X4: A] :
( Y3
= ( product_Pair @ A @ A @ X4 @ X4 ) ) ) ) ) ).
% closed_diagonal
thf(fact_5679_Bseq__add__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C3: A] :
( ( bfun @ nat @ A
@ ^ [X4: nat] : ( plus_plus @ A @ ( F3 @ X4 ) @ C3 )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ).
% Bseq_add_iff
thf(fact_5680_Bseq__add,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C3: A] :
( ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X4: nat] : ( plus_plus @ A @ ( F3 @ X4 ) @ C3 )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_add
thf(fact_5681_Bseq__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( bfun @ nat @ A
@ ^ [N3: nat] : ( F3 @ ( suc @ N3 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ).
% Bseq_Suc_iff
thf(fact_5682_Bseq__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,K2: nat] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [N3: nat] : ( X6 @ ( plus_plus @ nat @ N3 @ K2 ) )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_ignore_initial_segment
thf(fact_5683_Bseq__offset,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,K2: nat] :
( ( bfun @ nat @ A
@ ^ [N3: nat] : ( X6 @ ( plus_plus @ nat @ N3 @ K2 ) )
@ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) ) ) ) ).
% Bseq_offset
thf(fact_5684_closed__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo7761053866217962861closed @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y3: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y3 ) )
& ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% closed_superdiagonal
thf(fact_5685_closed__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo7761053866217962861closed @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y3: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y3 ) )
& ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ) ) ).
% closed_subdiagonal
thf(fact_5686_t4__space,axiom,
! [A: $tType] :
( ( topological_t4_space @ A )
=> ! [S3: set @ A,T4: set @ A] :
( ( topolo7761053866217962861closed @ A @ S3 )
=> ( ( topolo7761053866217962861closed @ A @ T4 )
=> ( ( ( inf_inf @ ( set @ A ) @ S3 @ T4 )
= ( bot_bot @ ( set @ A ) ) )
=> ? [U5: set @ A,V6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U5 )
& ( topolo1002775350975398744n_open @ A @ V6 )
& ( ord_less_eq @ ( set @ A ) @ S3 @ U5 )
& ( ord_less_eq @ ( set @ A ) @ T4 @ V6 )
& ( ( inf_inf @ ( set @ A ) @ U5 @ V6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% t4_space
thf(fact_5687_t3__space,axiom,
! [A: $tType] :
( ( topological_t3_space @ A )
=> ! [S3: set @ A,Y: A] :
( ( topolo7761053866217962861closed @ A @ S3 )
=> ( ~ ( member @ A @ Y @ S3 )
=> ? [U5: set @ A,V6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U5 )
& ( topolo1002775350975398744n_open @ A @ V6 )
& ( member @ A @ Y @ U5 )
& ( ord_less_eq @ ( set @ A ) @ S3 @ V6 )
& ( ( inf_inf @ ( set @ A ) @ U5 @ V6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% t3_space
thf(fact_5688_nhds__closed,axiom,
! [A: $tType] :
( ( topological_t3_space @ A )
=> ! [X3: A,A6: set @ A] :
( ( member @ A @ X3 @ A6 )
=> ( ( topolo1002775350975398744n_open @ A @ A6 )
=> ? [A9: set @ A] :
( ( member @ A @ X3 @ A9 )
& ( topolo7761053866217962861closed @ A @ A9 )
& ( ord_less_eq @ ( set @ A ) @ A9 @ A6 )
& ( eventually @ A
@ ^ [Y3: A] : ( member @ A @ Y3 @ A9 )
@ ( topolo7230453075368039082e_nhds @ A @ X3 ) ) ) ) ) ) ).
% nhds_closed
thf(fact_5689_Bseq__iff1a,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
= ( ? [N6: nat] :
! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( X6 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) ) ) ).
% Bseq_iff1a
thf(fact_5690_Bseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
= ( ? [N6: nat] :
! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X6 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) ) ) ).
% Bseq_iff
thf(fact_5691_Bseq__realpow,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( bfun @ nat @ real @ ( power_power @ real @ X3 ) @ ( at_top @ nat ) ) ) ) ).
% Bseq_realpow
thf(fact_5692_compact__fip,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo2193935891317330818ompact @ A )
= ( ^ [U6: set @ A] :
! [A7: set @ ( set @ A )] :
( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A7 )
=> ( topolo7761053866217962861closed @ A @ X4 ) )
=> ( ! [B6: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ B6 @ A7 )
=> ( ( finite_finite2 @ ( set @ A ) @ B6 )
=> ( ( inf_inf @ ( set @ A ) @ U6 @ ( complete_Inf_Inf @ ( set @ A ) @ B6 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ U6 @ ( complete_Inf_Inf @ ( set @ A ) @ A7 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% compact_fip
thf(fact_5693_compact__imp__fip,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,F6: set @ ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ! [T5: set @ A] :
( ( member @ ( set @ A ) @ T5 @ F6 )
=> ( topolo7761053866217962861closed @ A @ T5 ) )
=> ( ! [F16: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F16 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F16 @ F6 )
=> ( ( inf_inf @ ( set @ A ) @ S3 @ ( complete_Inf_Inf @ ( set @ A ) @ F16 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ S3 @ ( complete_Inf_Inf @ ( set @ A ) @ F6 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% compact_imp_fip
thf(fact_5694_ran__map__upd,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A3: B,B2: A] :
( ( ( M2 @ A3 )
= ( none @ A ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ A3 @ ( some @ A @ B2 ) ) )
= ( insert2 @ A @ B2 @ ( ran @ B @ A @ M2 ) ) ) ) ).
% ran_map_upd
thf(fact_5695_lenlex__append2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Us: list @ A,Xs2: list @ A,Ys: list @ A] :
( ( irrefl @ A @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Xs2 ) @ ( append @ A @ Us @ Ys ) ) @ ( lenlex @ A @ R ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( lenlex @ A @ R ) ) ) ) ).
% lenlex_append2
thf(fact_5696_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_5697_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_5698_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X4: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_5699_inj__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F3: A > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ A @ C @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_apfst
thf(fact_5700_inj__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F3: B > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ B @ C @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% inj_apsnd
thf(fact_5701_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( irrefl @ A @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys ) @ ( append @ A @ Xs2 @ Zs2 ) ) @ ( lexord @ A @ R2 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_5702_inj__on__insert,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: A,A6: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( insert2 @ A @ A3 @ A6 ) )
= ( ( inj_on @ A @ B @ F3 @ A6 )
& ~ ( member @ B @ ( F3 @ A3 ) @ ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_5703_inj__on__add_H,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A6: set @ A] :
( inj_on @ A @ A
@ ^ [B8: A] : ( plus_plus @ A @ B8 @ A3 )
@ A6 ) ) ).
% inj_on_add'
thf(fact_5704_inj__on__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A6: set @ A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A3 ) @ A6 ) ) ).
% inj_on_add
thf(fact_5705_inj__add__left,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_add_left
thf(fact_5706_option_Oinj__map,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( option @ A ) @ ( option @ B ) @ ( map_option @ A @ B @ F3 ) @ ( top_top @ ( set @ ( option @ A ) ) ) ) ) ).
% option.inj_map
thf(fact_5707_ranI,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A3: B,B2: A] :
( ( ( M2 @ A3 )
= ( some @ A @ B2 ) )
=> ( member @ A @ B2 @ ( ran @ B @ A @ M2 ) ) ) ).
% ranI
thf(fact_5708_irreflI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R )
=> ( irrefl @ A @ R ) ) ).
% irreflI
thf(fact_5709_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [A8: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ A8 ) @ R5 ) ) ) ).
% irrefl_def
thf(fact_5710_subset__inj__on,axiom,
! [B: $tType,A: $tType,F3: A > B,B5: set @ A,A6: set @ A] :
( ( inj_on @ A @ B @ F3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( inj_on @ A @ B @ F3 @ A6 ) ) ) ).
% subset_inj_on
thf(fact_5711_inj__on__subset,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( inj_on @ A @ B @ F3 @ B5 ) ) ) ).
% inj_on_subset
thf(fact_5712_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A6: set @ A,F3: A > B] :
( ! [X5: A,Y4: A] :
( ( ord_less @ A @ X5 @ Y4 )
=> ( ( member @ A @ X5 @ A6 )
=> ( ( member @ A @ Y4 @ A6 )
=> ( ( F3 @ X5 )
!= ( F3 @ Y4 ) ) ) ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ A6 )
=> ( ( member @ A @ Y4 @ A6 )
=> ( ( ord_less_eq @ A @ X5 @ Y4 )
| ( ord_less_eq @ A @ Y4 @ X5 ) ) ) )
=> ( inj_on @ A @ B @ F3 @ A6 ) ) ) ) ).
% linorder_inj_onI
thf(fact_5713_inj__img__insertE,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,X3: B,B5: set @ B] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ~ ( member @ B @ X3 @ B5 )
=> ( ( ( insert2 @ B @ X3 @ B5 )
= ( image2 @ A @ B @ F3 @ A6 ) )
=> ~ ! [X17: A,A9: set @ A] :
( ~ ( member @ A @ X17 @ A9 )
=> ( ( A6
= ( insert2 @ A @ X17 @ A9 ) )
=> ( ( X3
= ( F3 @ X17 ) )
=> ( B5
!= ( image2 @ A @ B @ F3 @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_5714_subset__image__inj,axiom,
! [A: $tType,B: $tType,S3: set @ A,F3: B > A,T4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ ( image2 @ B @ A @ F3 @ T4 ) )
= ( ? [U6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U6 @ T4 )
& ( inj_on @ B @ A @ F3 @ U6 )
& ( S3
= ( image2 @ B @ A @ F3 @ U6 ) ) ) ) ) ).
% subset_image_inj
thf(fact_5715_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,B5: set @ A,A3: A,A6: set @ A] :
( ( inj_on @ A @ B @ F3 @ B5 )
=> ( ( member @ A @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( member @ B @ ( F3 @ A3 ) @ ( image2 @ A @ B @ F3 @ A6 ) )
= ( member @ A @ A3 @ A6 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_5716_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,C4: set @ A,A6: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ( ( image2 @ A @ B @ F3 @ A6 )
= ( image2 @ A @ B @ F3 @ B5 ) )
= ( A6 = B5 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_5717_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_5718_irrefl__distinct,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [A8: A,B8: A] : A8 != B8
@ X4 ) ) ) ) ).
% irrefl_distinct
thf(fact_5719_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A6 ) @ ( image2 @ A @ B @ F3 @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A6 @ B5 ) ) ) ).
% inj_image_subset_iff
thf(fact_5720_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A6: set @ A,A11: set @ B] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A6 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A6 ) @ A11 ) ) )
= ( ? [G4: B > A] :
( ( image2 @ B @ A @ G4 @ A11 )
= A6 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_5721_finite__surj__inj,axiom,
! [A: $tType,A6: set @ A,F3: A > A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( image2 @ A @ A @ F3 @ A6 ) )
=> ( inj_on @ A @ A @ F3 @ A6 ) ) ) ).
% finite_surj_inj
thf(fact_5722_inj__on__finite,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,B5: set @ B] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A6 ) @ B5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( finite_finite2 @ A @ A6 ) ) ) ) ).
% inj_on_finite
thf(fact_5723_endo__inj__surj,axiom,
! [A: $tType,A6: set @ A,F3: A > A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ A @ A @ F3 @ A6 ) @ A6 )
=> ( ( inj_on @ A @ A @ F3 @ A6 )
=> ( ( image2 @ A @ A @ F3 @ A6 )
= A6 ) ) ) ) ).
% endo_inj_surj
thf(fact_5724_inj__on__image__Int,axiom,
! [B: $tType,A: $tType,F3: A > B,C4: set @ A,A6: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ( image2 @ A @ B @ F3 @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) )
= ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A6 ) @ ( image2 @ A @ B @ F3 @ B5 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_5725_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F3: A > B,C4: set @ A,A6: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) )
= ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A6 ) @ ( image2 @ A @ B @ F3 @ B5 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_5726_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,X3: B,B5: set @ A] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ( member @ B @ X3 @ ( image2 @ A @ B @ F3 @ A6 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( member @ A @ ( the_inv_into @ A @ B @ A6 @ F3 @ X3 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_5727_ran__restrictD,axiom,
! [B: $tType,A: $tType,Y: A,M2: B > ( option @ A ),A6: set @ B] :
( ( member @ A @ Y @ ( ran @ B @ A @ ( restrict_map @ B @ A @ M2 @ A6 ) ) )
=> ? [X5: B] :
( ( member @ B @ X5 @ A6 )
& ( ( M2 @ X5 )
= ( some @ A @ Y ) ) ) ) ).
% ran_restrictD
thf(fact_5728_inj__on__UNION__chain,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,A6: A > ( set @ B ),F3: B > C] :
( ! [I3: A,J2: A] :
( ( member @ A @ I3 @ I5 )
=> ( ( member @ A @ J2 @ I5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A6 @ I3 ) @ ( A6 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A6 @ J2 ) @ ( A6 @ I3 ) ) ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( inj_on @ B @ C @ F3 @ ( A6 @ I3 ) ) )
=> ( inj_on @ B @ C @ F3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ I5 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_5729_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,F3: B > C,A6: A > ( set @ B )] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( inj_on @ B @ C @ F3 @ ( A6 @ I3 ) ) )
=> ( inj_on @ B @ C @ F3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ I5 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_5730_ran__def,axiom,
! [B: $tType,A: $tType] :
( ( ran @ A @ B )
= ( ^ [M5: A > ( option @ B )] :
( collect @ B
@ ^ [B8: B] :
? [A8: A] :
( ( M5 @ A8 )
= ( some @ B @ B8 ) ) ) ) ) ).
% ran_def
thf(fact_5731_surjective__iff__injective__gen,axiom,
! [B: $tType,A: $tType,S3: set @ A,T4: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ( ( finite_card @ A @ S3 )
= ( finite_card @ B @ T4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ S3 ) @ T4 )
=> ( ( ! [X4: B] :
( ( member @ B @ X4 @ T4 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ S3 )
& ( ( F3 @ Y3 )
= X4 ) ) ) )
= ( inj_on @ A @ B @ F3 @ S3 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_5732_card__bij__eq,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ A,B5: set @ B,G3: B > A] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A6 ) @ B5 )
=> ( ( inj_on @ B @ A @ G3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G3 @ B5 ) @ A6 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( finite_card @ A @ A6 )
= ( finite_card @ B @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_5733_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A6 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_5734_lexl__not__refl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: list @ A] :
( ( irrefl @ A @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ X3 ) @ ( lex @ A @ R2 ) ) ) ).
% lexl_not_refl
thf(fact_5735_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,G3: A > B,B5: set @ A] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ( inj_on @ A @ B @ G3 @ B5 )
=> ( ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A6 ) @ ( image2 @ A @ B @ G3 @ B5 ) )
= ( bot_bot @ ( set @ B ) ) )
=> ( inj_on @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ A6 ) @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_5736_image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > B,C4: set @ A,A6: set @ C,B5: C > ( set @ A ),J: C] :
( ( inj_on @ A @ B @ F3 @ C4 )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ ( B5 @ X5 ) @ C4 ) )
=> ( ( member @ C @ J @ A6 )
=> ( ( image2 @ A @ B @ F3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B5 @ A6 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X4: C] : ( image2 @ A @ B @ F3 @ ( B5 @ X4 ) )
@ A6 ) ) ) ) ) ) ).
% image_INT
thf(fact_5737_inj__on__iff__card__le,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A6 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A6 ) @ B5 ) ) )
= ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B5 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_5738_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ A,B5: set @ B] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A6 ) @ B5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B5 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_5739_card__le__inj,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B5 ) )
=> ? [F2: A > B] :
( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A6 ) @ B5 )
& ( inj_on @ A @ B @ F2 @ A6 ) ) ) ) ) ).
% card_le_inj
thf(fact_5740_inj__on__Un,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( ( inj_on @ A @ B @ F3 @ A6 )
& ( inj_on @ A @ B @ F3 @ B5 )
& ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A6 @ B5 ) ) @ ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ B5 @ A6 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% inj_on_Un
thf(fact_5741_Schroeder__Bernstein,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ A,B5: set @ B,G3: B > A] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A6 ) @ B5 )
=> ( ( inj_on @ B @ A @ G3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G3 @ B5 ) @ A6 )
=> ? [H4: A > B] : ( bij_betw @ A @ B @ H4 @ A6 @ B5 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_5742_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G3: A > B,C4: set @ A,B5: set @ A,X3: A] :
( ( inj_on @ A @ B @ G3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ C4 @ ( sup_sup @ ( set @ A ) @ B5 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ ( B > A )
@ ^ [I4: B] : ( if @ A @ ( member @ B @ I4 @ ( image2 @ A @ B @ G3 @ C4 ) ) @ ( the_inv_into @ A @ B @ C4 @ G3 @ I4 ) @ X3 )
@ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B5 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_5743_all__subset__image__inj,axiom,
! [A: $tType,B: $tType,F3: B > A,S3: set @ B,P: ( set @ A ) > $o] :
( ( ! [T9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T9 @ ( image2 @ B @ A @ F3 @ S3 ) )
=> ( P @ T9 ) ) )
= ( ! [T9: set @ B] :
( ( ( ord_less_eq @ ( set @ B ) @ T9 @ S3 )
& ( inj_on @ B @ A @ F3 @ T9 ) )
=> ( P @ ( image2 @ B @ A @ F3 @ T9 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_5744_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_5745_inj__Suc,axiom,
! [N5: set @ nat] : ( inj_on @ nat @ nat @ suc @ N5 ) ).
% inj_Suc
thf(fact_5746_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F3: A > B,X6: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ ( F3 @ X4 ) )
@ X6 ) ).
% inj_on_convol_ident
thf(fact_5747_inj__Some,axiom,
! [A: $tType,A6: set @ A] : ( inj_on @ A @ ( option @ A ) @ ( some @ A ) @ A6 ) ).
% inj_Some
thf(fact_5748_inj__singleton,axiom,
! [A: $tType,A6: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X4: A] : ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
@ A6 ) ).
% inj_singleton
thf(fact_5749_inj__on__diff__nat,axiom,
! [N5: set @ nat,K2: nat] :
( ! [N2: nat] :
( ( member @ nat @ N2 @ N5 )
=> ( ord_less_eq @ nat @ K2 @ N2 ) )
=> ( inj_on @ nat @ nat
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ K2 )
@ N5 ) ) ).
% inj_on_diff_nat
thf(fact_5750_swap__inj__on,axiom,
! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B )] :
( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [I4: A,J3: B] : ( product_Pair @ B @ A @ J3 @ I4 ) )
@ A6 ) ).
% swap_inj_on
thf(fact_5751_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_5752_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X8: $o > A,Y10: $o > A] :
( ( ord_less_eq @ A @ ( X8 @ $false ) @ ( Y10 @ $false ) )
& ( ord_less_eq @ A @ ( X8 @ $true ) @ ( Y10 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_5753_inj__on__nth,axiom,
! [A: $tType,Xs2: list @ A,I5: set @ nat] :
( ( distinct @ A @ Xs2 )
=> ( ! [X5: nat] :
( ( member @ nat @ X5 @ I5 )
=> ( ord_less @ nat @ X5 @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
=> ( inj_on @ nat @ A @ ( nth @ A @ Xs2 ) @ I5 ) ) ) ).
% inj_on_nth
thf(fact_5754_infinite__iff__countable__subset,axiom,
! [A: $tType,S3: set @ A] :
( ( ~ ( finite_finite2 @ A @ S3 ) )
= ( ? [F4: nat > A] :
( ( inj_on @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_5755_infinite__countable__subset,axiom,
! [A: $tType,S3: set @ A] :
( ~ ( finite_finite2 @ A @ S3 )
=> ? [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 ) ) ) @ S3 ) ) ) ).
% infinite_countable_subset
thf(fact_5756_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_5757_ex__subset__image__inj,axiom,
! [A: $tType,B: $tType,F3: B > A,S3: set @ B,P: ( set @ A ) > $o] :
( ( ? [T9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T9 @ ( image2 @ B @ A @ F3 @ S3 ) )
& ( P @ T9 ) ) )
= ( ? [T9: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ T9 @ S3 )
& ( inj_on @ B @ A @ F3 @ T9 )
& ( P @ ( image2 @ B @ A @ F3 @ T9 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_5758_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: B > A,A19: set @ B,B14: set @ A,F22: C > D,B23: set @ C,A25: set @ D] :
( ( ( image2 @ B @ A @ F1 @ A19 )
= B14 )
=> ( ( inj_on @ C @ D @ F22 @ B23 )
=> ( ( ord_less_eq @ ( set @ D ) @ ( image2 @ C @ D @ F22 @ B23 ) @ A25 )
=> ( ( ( B23
= ( bot_bot @ ( set @ C ) ) )
=> ( A25
= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( bNF_Wellorder_Func @ C @ A @ B23 @ B14 )
= ( image2 @ ( D > B ) @ ( C > A ) @ ( bNF_We4925052301507509544nc_map @ C @ B @ A @ D @ B23 @ F1 @ F22 ) @ ( bNF_Wellorder_Func @ D @ B @ A25 @ A19 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_5759_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),X3: B,Y: A,Z2: A] :
( ( ( M2 @ X3 )
= ( some @ A @ Y ) )
=> ( ( inj_on @ B @ ( option @ A ) @ M2 @ ( dom @ B @ A @ M2 ) )
=> ( ~ ( member @ A @ Z2 @ ( ran @ B @ A @ M2 ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ X3 @ ( some @ A @ Z2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M2 ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_5760_Func__non__emp,axiom,
! [A: $tType,B: $tType,B5: set @ A,A6: set @ B] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bNF_Wellorder_Func @ B @ A @ A6 @ B5 )
!= ( bot_bot @ ( set @ ( B > A ) ) ) ) ) ).
% Func_non_emp
thf(fact_5761_dom__eq__empty__conv,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B )] :
( ( ( dom @ A @ B @ F3 )
= ( bot_bot @ ( set @ A ) ) )
= ( F3
= ( ^ [X4: A] : ( none @ B ) ) ) ) ).
% dom_eq_empty_conv
thf(fact_5762_fun__upd__None__if__notin__dom,axiom,
! [B: $tType,A: $tType,K2: A,M2: A > ( option @ B )] :
( ~ ( member @ A @ K2 @ ( dom @ A @ B @ M2 ) )
=> ( ( fun_upd @ A @ ( option @ B ) @ M2 @ K2 @ ( none @ B ) )
= M2 ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_5763_dom__const,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( dom @ A @ B
@ ^ [X4: A] : ( some @ B @ ( F3 @ X4 ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% dom_const
thf(fact_5764_dom__empty,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B
@ ^ [X4: A] : ( none @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% dom_empty
thf(fact_5765_dom__fun__upd,axiom,
! [B: $tType,A: $tType,Y: option @ B,F3: A > ( option @ B ),X3: A] :
( ( ( Y
= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X3 @ Y ) )
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F3 ) @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( ( Y
!= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X3 @ Y ) )
= ( insert2 @ A @ X3 @ ( dom @ A @ B @ F3 ) ) ) ) ) ).
% dom_fun_upd
thf(fact_5766_dom__def,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B )
= ( ^ [M5: A > ( option @ B )] :
( collect @ A
@ ^ [A8: A] :
( ( M5 @ A8 )
!= ( none @ B ) ) ) ) ) ).
% dom_def
thf(fact_5767_domIff,axiom,
! [A: $tType,B: $tType,A3: A,M2: A > ( option @ B )] :
( ( member @ A @ A3 @ ( dom @ A @ B @ M2 ) )
= ( ( M2 @ A3 )
!= ( none @ B ) ) ) ).
% domIff
thf(fact_5768_domD,axiom,
! [A: $tType,B: $tType,A3: A,M2: A > ( option @ B )] :
( ( member @ A @ A3 @ ( dom @ A @ B @ M2 ) )
=> ? [B4: B] :
( ( M2 @ A3 )
= ( some @ B @ B4 ) ) ) ).
% domD
thf(fact_5769_domI,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A3: B,B2: A] :
( ( ( M2 @ A3 )
= ( some @ A @ B2 ) )
=> ( member @ B @ A3 @ ( dom @ B @ A @ M2 ) ) ) ).
% domI
thf(fact_5770_insert__dom,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),X3: B,Y: A] :
( ( ( F3 @ X3 )
= ( some @ A @ Y ) )
=> ( ( insert2 @ B @ X3 @ ( dom @ B @ A @ F3 ) )
= ( dom @ B @ A @ F3 ) ) ) ).
% insert_dom
thf(fact_5771_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 ) ) )
=> ? [X5: A] :
( ( F3 @ X5 )
= ( none @ B ) ) ) ) ).
% finite_map_freshness
thf(fact_5772_dom__minus,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),X3: B,A6: set @ B] :
( ( ( F3 @ X3 )
= ( none @ A ) )
=> ( ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F3 ) @ ( insert2 @ B @ X3 @ A6 ) )
= ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F3 ) @ A6 ) ) ) ).
% dom_minus
thf(fact_5773_finite__set__of__finite__maps,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( finite_finite2 @ ( A > ( option @ B ) )
@ ( collect @ ( A > ( option @ B ) )
@ ^ [M5: A > ( option @ B )] :
( ( ( dom @ A @ B @ M5 )
= A6 )
& ( ord_less_eq @ ( set @ B ) @ ( ran @ A @ B @ M5 ) @ B5 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_5774_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M5: A > ( option @ B )] :
( image2 @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ ( the2 @ B @ ( M5 @ X4 ) ) )
@ ( dom @ A @ B @ M5 ) ) ) ) ).
% graph_eq_to_snd_dom
thf(fact_5775_finite__Map__induct,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),P: ( A > ( option @ B ) ) > $o] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M2 ) )
=> ( ( P
@ ^ [X4: A] : ( none @ B ) )
=> ( ! [K: A,V: B,M: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M ) )
=> ( ~ ( member @ A @ K @ ( dom @ A @ B @ M ) )
=> ( ( P @ M )
=> ( P @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( some @ B @ V ) ) ) ) ) )
=> ( P @ M2 ) ) ) ) ).
% finite_Map_induct
thf(fact_5776_Func__map,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,G3: A > B,A25: set @ A,A19: set @ B,F1: B > C,B14: set @ C,F22: D > A,B23: set @ D] :
( ( member @ ( A > B ) @ G3 @ ( bNF_Wellorder_Func @ A @ B @ A25 @ A19 ) )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ F1 @ A19 ) @ B14 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ D @ A @ F22 @ B23 ) @ A25 )
=> ( member @ ( D > C ) @ ( bNF_We4925052301507509544nc_map @ D @ B @ C @ A @ B23 @ F1 @ F22 @ G3 ) @ ( bNF_Wellorder_Func @ D @ C @ B23 @ B14 ) ) ) ) ) ).
% Func_map
thf(fact_5777_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B ),X3: A] :
( ( ( dom @ A @ B @ F3 )
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ? [V5: B] :
( F3
= ( fun_upd @ A @ ( option @ B )
@ ^ [X4: A] : ( none @ B )
@ X3
@ ( some @ B @ V5 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_5778_Func__is__emp,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: set @ B] :
( ( ( bNF_Wellorder_Func @ A @ B @ A6 @ B5 )
= ( bot_bot @ ( set @ ( A > B ) ) ) )
= ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Func_is_emp
thf(fact_5779_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 ) )
@ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R )
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ).
% rtrancl_finite_eq_relpow
thf(fact_5780_cclfp__def,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( order_532582986084564980_cclfp @ A )
= ( ^ [F4: A > A] :
( complete_Sup_Sup @ A
@ ( image2 @ nat @ A
@ ^ [I4: nat] : ( compow @ ( A > A ) @ I4 @ F4 @ ( bot_bot @ A ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% cclfp_def
thf(fact_5781_set__list__bind,axiom,
! [A: $tType,B: $tType,Xs2: list @ B,F3: B > ( list @ A )] :
( ( set2 @ A @ ( bind @ B @ A @ Xs2 @ F3 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( set2 @ A @ ( F3 @ X4 ) )
@ ( set2 @ B @ Xs2 ) ) ) ) ).
% set_list_bind
thf(fact_5782_trancl__rtrancl__trancl,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_5783_rtrancl__trancl__trancl,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_5784_rtrancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_5785_rtrancl__into__trancl1,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_5786_rtrancl__eq__or__trancl,axiom,
! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R ) )
= ( ( X3 = Y )
| ( ( X3 != Y )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_5787_trancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ).
% trancl_into_rtrancl
thf(fact_5788_tranclD2,axiom,
! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R ) )
=> ? [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z3 ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ R ) ) ) ).
% tranclD2
thf(fact_5789_rtranclD,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( A3 = B2 )
| ( ( A3 != B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% rtranclD
thf(fact_5790_tranclD,axiom,
! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R ) )
=> ? [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z3 ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% tranclD
thf(fact_5791_rtrancl__Un__separatorE,axiom,
! [A: $tType,A3: A,B2: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X5 ) @ ( transitive_rtrancl @ A @ P ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ Q )
=> ( X5 = Y4 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_5792_rtrancl__Un__separator__converseE,axiom,
! [A: $tType,A3: A,B2: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ B2 ) @ ( transitive_rtrancl @ A @ P ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X5 ) @ Q )
=> ( Y4 = X5 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_5793_rtrancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( A22 != A1 )
=> ~ ! [B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B4 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A22 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_5794_rtrancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R2 ) )
= ( ? [A8: A] :
( ( A1 = A8 )
& ( A22 = A8 ) )
| ? [A8: A,B8: A,C6: A] :
( ( A1 = A8 )
& ( A22 = C6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B8 ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ C6 ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_5795_rtrancl_Ortrancl__refl,axiom,
! [A: $tType,A3: A,R2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_5796_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_5797_rtranclE,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( A3 != B2 )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ B2 ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_5798_rtrancl__trans,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_5799_rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( P @ A3 )
=> ( ! [Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 )
=> ( ( P @ Y4 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% rtrancl_induct
thf(fact_5800_converse__rtranclE,axiom,
! [A: $tType,X3: A,Z2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( X3 != Z2 )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_5801_converse__rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( P @ B2 )
=> ( ! [Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( P @ Z3 )
=> ( P @ Y4 ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_5802_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_5803_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_5804_converse__rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( P @ Bx @ By )
=> ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( P @ Aa2 @ Ba )
=> ( P @ A5 @ B4 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_5805_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa2: A,Xb: B,Za: A,Zb: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa2 @ Xb ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( ( product_Pair @ A @ B @ Xa2 @ Xb )
!= ( product_Pair @ A @ B @ Za @ Zb ) )
=> ~ ! [A5: A,B4: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa2 @ Xb ) @ ( product_Pair @ A @ B @ A5 @ B4 ) ) @ R2 )
=> ~ ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_5806_rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B4 ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 )
=> ( ( P @ A5 @ B4 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_5807_list__bind__cong,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ A,F3: A > ( list @ B ),G3: A > ( list @ B )] :
( ( Xs2 = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ( F3 @ X5 )
= ( G3 @ X5 ) ) )
=> ( ( bind @ A @ B @ Xs2 @ F3 )
= ( bind @ A @ B @ Ys @ G3 ) ) ) ) ).
% list_bind_cong
thf(fact_5808_rtrancl__listrel1__ConsI1,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),X3: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ X3 @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ).
% rtrancl_listrel1_ConsI1
thf(fact_5809_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) )
=> ( ( size_size @ ( list @ A ) @ X3 )
= ( size_size @ ( list @ A ) @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_5810_pred__nat__trancl__eq__le,axiom,
! [M2: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M2 @ N ) @ ( transitive_rtrancl @ nat @ pred_nat ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% pred_nat_trancl_eq_le
thf(fact_5811_rtrancl__insert,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R2 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ A3 ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% rtrancl_insert
thf(fact_5812_trancl__insert,axiom,
! [A: $tType,Y: A,X3: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X3 ) @ R2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A8: A,B8: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B8 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% trancl_insert
thf(fact_5813_set__nths,axiom,
! [A: $tType,Xs2: list @ A,I5: set @ nat] :
( ( set2 @ A @ ( nths @ A @ Xs2 @ I5 ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [I4: nat] :
( ( Uu3
= ( nth @ A @ Xs2 @ I4 ) )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( member @ nat @ I4 @ I5 ) ) ) ) ).
% set_nths
thf(fact_5814_finite__subsets__at__top__finite,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite5375528669736107172at_top @ A @ A6 )
= ( principal @ ( set @ A ) @ ( insert2 @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ).
% finite_subsets_at_top_finite
thf(fact_5815_comp__fun__idem__on_Ofold__insert__idem,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X3: A,A6: set @ A,Z2: B] :
( ( finite673082921795544331dem_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X3 @ A6 ) @ S3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( insert2 @ A @ X3 @ A6 ) )
= ( F3 @ X3 @ ( finite_fold @ A @ B @ F3 @ Z2 @ A6 ) ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem
thf(fact_5816_eventually__finite__subsets__at__top__weakI,axiom,
! [A: $tType,A6: set @ A,P: ( set @ A ) > $o] :
( ! [X10: set @ A] :
( ( finite_finite2 @ A @ X10 )
=> ( ( ord_less_eq @ ( set @ A ) @ X10 @ A6 )
=> ( P @ X10 ) ) )
=> ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A6 ) ) ) ).
% eventually_finite_subsets_at_top_weakI
thf(fact_5817_nths__empty,axiom,
! [A: $tType,Xs2: list @ A] :
( ( nths @ A @ Xs2 @ ( bot_bot @ ( set @ nat ) ) )
= ( nil @ A ) ) ).
% nths_empty
thf(fact_5818_notin__set__nthsI,axiom,
! [A: $tType,X3: A,Xs2: list @ A,I5: set @ nat] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ~ ( member @ A @ X3 @ ( set2 @ A @ ( nths @ A @ Xs2 @ I5 ) ) ) ) ).
% notin_set_nthsI
thf(fact_5819_in__set__nthsD,axiom,
! [A: $tType,X3: A,Xs2: list @ A,I5: set @ nat] :
( ( member @ A @ X3 @ ( set2 @ A @ ( nths @ A @ Xs2 @ I5 ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) ) ) ).
% in_set_nthsD
thf(fact_5820_finite__subsets__at__top__neq__bot,axiom,
! [A: $tType,A6: set @ A] :
( ( finite5375528669736107172at_top @ A @ A6 )
!= ( bot_bot @ ( filter @ ( set @ A ) ) ) ) ).
% finite_subsets_at_top_neq_bot
thf(fact_5821_eventually__finite__subsets__at__top__finite,axiom,
! [A: $tType,A6: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A6 )
=> ( ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A6 ) )
= ( P @ A6 ) ) ) ).
% eventually_finite_subsets_at_top_finite
thf(fact_5822_set__nths__subset,axiom,
! [A: $tType,Xs2: list @ A,I5: set @ nat] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( nths @ A @ Xs2 @ I5 ) ) @ ( set2 @ A @ Xs2 ) ) ).
% set_nths_subset
thf(fact_5823_nths__all,axiom,
! [A: $tType,Xs2: list @ A,I5: set @ nat] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( member @ nat @ I3 @ I5 ) )
=> ( ( nths @ A @ Xs2 @ I5 )
= Xs2 ) ) ).
% nths_all
thf(fact_5824_eventually__finite__subsets__at__top,axiom,
! [A: $tType,P: ( set @ A ) > $o,A6: set @ A] :
( ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A6 ) )
= ( ? [X8: set @ A] :
( ( finite_finite2 @ A @ X8 )
& ( ord_less_eq @ ( set @ A ) @ X8 @ A6 )
& ! [Y10: set @ A] :
( ( ( finite_finite2 @ A @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ X8 @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ Y10 @ A6 ) )
=> ( P @ Y10 ) ) ) ) ) ).
% eventually_finite_subsets_at_top
thf(fact_5825_nths__append,axiom,
! [A: $tType,L: list @ A,L3: list @ A,A6: set @ nat] :
( ( nths @ A @ ( append @ A @ L @ L3 ) @ A6 )
= ( append @ A @ ( nths @ A @ L @ A6 )
@ ( nths @ A @ L3
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( plus_plus @ nat @ J3 @ ( size_size @ ( list @ A ) @ L ) ) @ A6 ) ) ) ) ) ).
% nths_append
thf(fact_5826_length__nths,axiom,
! [A: $tType,Xs2: list @ A,I5: set @ nat] :
( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs2 @ I5 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( member @ nat @ I4 @ I5 ) ) ) ) ) ).
% length_nths
thf(fact_5827_finite__subsets__at__top__def,axiom,
! [A: $tType] :
( ( finite5375528669736107172at_top @ A )
= ( ^ [A7: set @ A] :
( complete_Inf_Inf @ ( filter @ ( set @ A ) )
@ ( image2 @ ( set @ A ) @ ( filter @ ( set @ A ) )
@ ^ [X8: set @ A] :
( principal @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Y10: set @ A] :
( ( finite_finite2 @ A @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ X8 @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ Y10 @ A7 ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [X8: set @ A] :
( ( finite_finite2 @ A @ X8 )
& ( ord_less_eq @ ( set @ A ) @ X8 @ A7 ) ) ) ) ) ) ) ).
% finite_subsets_at_top_def
thf(fact_5828_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_5829_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_5830_filterlim__finite__subsets__at__top,axiom,
! [A: $tType,B: $tType,F3: A > ( set @ B ),A6: set @ B,F6: filter @ A] :
( ( filterlim @ A @ ( set @ B ) @ F3 @ ( finite5375528669736107172at_top @ B @ A6 ) @ F6 )
= ( ! [X8: set @ B] :
( ( ( finite_finite2 @ B @ X8 )
& ( ord_less_eq @ ( set @ B ) @ X8 @ A6 ) )
=> ( eventually @ A
@ ^ [Y3: A] :
( ( finite_finite2 @ B @ ( F3 @ Y3 ) )
& ( ord_less_eq @ ( set @ B ) @ X8 @ ( F3 @ Y3 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F3 @ Y3 ) @ A6 ) )
@ F6 ) ) ) ) ).
% filterlim_finite_subsets_at_top
thf(fact_5831_nths__Cons,axiom,
! [A: $tType,X3: A,L: list @ A,A6: set @ nat] :
( ( nths @ A @ ( cons @ A @ X3 @ L ) @ A6 )
= ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A6 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( nil @ A ) )
@ ( nths @ A @ L
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( suc @ J3 ) @ A6 ) ) ) ) ) ).
% nths_Cons
thf(fact_5832_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
! [B: $tType,A: $tType,C: $tType,S3: set @ A,F3: A > B > B,G3: C > A,R: set @ C] :
( ( finite673082921795544331dem_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ G3 @ ( top_top @ ( set @ C ) ) ) @ S3 )
=> ( finite673082921795544331dem_on @ C @ B @ R @ ( comp @ A @ ( B > B ) @ C @ F3 @ G3 ) ) ) ) ).
% comp_fun_idem_on.comp_comp_fun_idem_on
thf(fact_5833_comp__fun__idem__on_Ofold__insert__idem2,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X3: A,A6: set @ A,Z2: B] :
( ( finite673082921795544331dem_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X3 @ A6 ) @ S3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( insert2 @ A @ X3 @ A6 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X3 @ Z2 ) @ A6 ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem2
thf(fact_5834_Sup__filter__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( filter @ A ) )
= ( ^ [S6: set @ ( filter @ A )] :
( abs_filter @ A
@ ^ [P4: A > $o] :
! [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ S6 )
=> ( eventually @ A @ P4 @ X4 ) ) ) ) ) ).
% Sup_filter_def
thf(fact_5835_ntrancl__Suc,axiom,
! [A: $tType,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( suc @ N ) @ R )
= ( relcomp @ A @ A @ A @ ( transitive_ntrancl @ A @ N @ R ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ R ) ) ) ).
% ntrancl_Suc
thf(fact_5836_nth__image,axiom,
! [A: $tType,L: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ L @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( image2 @ nat @ A @ ( nth @ A @ Xs2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ L ) )
= ( set2 @ A @ ( take @ A @ L @ Xs2 ) ) ) ) ).
% nth_image
thf(fact_5837_IdI,axiom,
! [A: $tType,A3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id2 @ A ) ) ).
% IdI
thf(fact_5838_pair__in__Id__conv,axiom,
! [A: $tType,A3: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id2 @ A ) )
= ( A3 = B2 ) ) ).
% pair_in_Id_conv
thf(fact_5839_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_5840_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_5841_take__Suc__Cons,axiom,
! [A: $tType,N: nat,X3: A,Xs2: list @ A] :
( ( take @ A @ ( suc @ N ) @ ( cons @ A @ X3 @ Xs2 ) )
= ( cons @ A @ X3 @ ( take @ A @ N @ Xs2 ) ) ) ).
% take_Suc_Cons
thf(fact_5842_take__all__iff,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( take @ A @ N @ Xs2 )
= Xs2 )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ).
% take_all_iff
thf(fact_5843_take__all,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N )
=> ( ( take @ A @ N @ Xs2 )
= Xs2 ) ) ).
% take_all
thf(fact_5844_take__update__cancel,axiom,
! [A: $tType,N: nat,M2: nat,Xs2: list @ A,Y: A] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( take @ A @ N @ ( list_update @ A @ Xs2 @ M2 @ Y ) )
= ( take @ A @ N @ Xs2 ) ) ) ).
% take_update_cancel
thf(fact_5845_take__append,axiom,
! [A: $tType,N: nat,Xs2: list @ A,Ys: list @ A] :
( ( take @ A @ N @ ( append @ A @ Xs2 @ Ys ) )
= ( append @ A @ ( take @ A @ N @ Xs2 ) @ ( take @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ Ys ) ) ) ).
% take_append
thf(fact_5846_take__Cons__numeral,axiom,
! [A: $tType,V2: num,X3: A,Xs2: list @ A] :
( ( take @ A @ ( numeral_numeral @ nat @ V2 ) @ ( cons @ A @ X3 @ Xs2 ) )
= ( cons @ A @ X3 @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) @ Xs2 ) ) ) ).
% take_Cons_numeral
thf(fact_5847_dom__map__upds,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),Xs2: list @ A,Ys: list @ B] :
( ( dom @ A @ B @ ( map_upds @ A @ B @ M2 @ Xs2 @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs2 ) ) @ ( dom @ A @ B @ M2 ) ) ) ).
% dom_map_upds
thf(fact_5848_bot__filter__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P4: A > $o] : $true ) ) ).
% bot_filter_def
thf(fact_5849_set__take__subset,axiom,
! [A: $tType,N: nat,Xs2: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ N @ Xs2 ) ) @ ( set2 @ A @ Xs2 ) ) ).
% set_take_subset
thf(fact_5850_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_5851_IdE,axiom,
! [A: $tType,P2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P2 @ ( id2 @ A ) )
=> ~ ! [X5: A] :
( P2
!= ( product_Pair @ A @ A @ X5 @ X5 ) ) ) ).
% IdE
thf(fact_5852_IdD,axiom,
! [A: $tType,A3: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id2 @ A ) )
=> ( A3 = B2 ) ) ).
% IdD
thf(fact_5853_in__set__takeD,axiom,
! [A: $tType,X3: A,N: nat,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( take @ A @ N @ Xs2 ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) ) ) ).
% in_set_takeD
thf(fact_5854_Id__def,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [P5: product_prod @ A @ A] :
? [X4: A] :
( P5
= ( product_Pair @ A @ A @ X4 @ X4 ) ) ) ) ).
% Id_def
thf(fact_5855_set__take__subset__set__take,axiom,
! [A: $tType,M2: nat,N: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ M2 @ Xs2 ) ) @ ( set2 @ A @ ( take @ A @ N @ Xs2 ) ) ) ) ).
% set_take_subset_set_take
thf(fact_5856_Id__fstsnd__eq,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [X4: product_prod @ A @ A] :
( ( product_fst @ A @ A @ X4 )
= ( product_snd @ A @ A @ X4 ) ) ) ) ).
% Id_fstsnd_eq
thf(fact_5857_sup__filter__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( filter @ A ) )
= ( ^ [F9: filter @ A,F10: filter @ A] :
( abs_filter @ A
@ ^ [P4: A > $o] :
( ( eventually @ A @ P4 @ F9 )
& ( eventually @ A @ P4 @ F10 ) ) ) ) ) ).
% sup_filter_def
thf(fact_5858_nth__take__lemma,axiom,
! [A: $tType,K2: nat,Xs2: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ K2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ K2 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K2 )
=> ( ( nth @ A @ Xs2 @ I3 )
= ( nth @ A @ Ys @ I3 ) ) )
=> ( ( take @ A @ K2 @ Xs2 )
= ( take @ A @ K2 @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_5859_principal__def,axiom,
! [A: $tType] :
( ( principal @ A )
= ( ^ [S6: set @ A] : ( abs_filter @ A @ ( ball @ A @ S6 ) ) ) ) ).
% principal_def
thf(fact_5860_reflcl__set__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( A > A > $o )
@ ^ [X4: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 )
@ ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [X4: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ).
% reflcl_set_eq
thf(fact_5861_map__filter__on__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter_on @ A @ B )
= ( ^ [X8: set @ A,F4: A > B,F9: filter @ A] :
( abs_filter @ B
@ ^ [P4: B > $o] :
( eventually @ A
@ ^ [X4: A] :
( ( P4 @ ( F4 @ X4 ) )
& ( member @ A @ X4 @ X8 ) )
@ F9 ) ) ) ) ).
% map_filter_on_def
thf(fact_5862_top__filter__def,axiom,
! [A: $tType] :
( ( top_top @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P3: A > $o] :
! [X7: A] : ( P3 @ X7 ) ) ) ).
% top_filter_def
thf(fact_5863_map__upd__upds__conv__if,axiom,
! [A: $tType,B: $tType,X3: A,Ys: list @ B,Xs2: list @ A,F3: A > ( option @ B ),Y: B] :
( ( ( member @ A @ X3 @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs2 ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X3 @ ( some @ B @ Y ) ) @ Xs2 @ Ys )
= ( map_upds @ A @ B @ F3 @ Xs2 @ Ys ) ) )
& ( ~ ( member @ A @ X3 @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs2 ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X3 @ ( some @ B @ Y ) ) @ Xs2 @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ F3 @ Xs2 @ Ys ) @ X3 @ ( some @ B @ Y ) ) ) ) ) ).
% map_upd_upds_conv_if
thf(fact_5864_filtercomap__def,axiom,
! [B: $tType,A: $tType] :
( ( filtercomap @ A @ B )
= ( ^ [F4: A > B,F9: filter @ B] :
( abs_filter @ A
@ ^ [P4: A > $o] :
? [Q6: B > $o] :
( ( eventually @ B @ Q6 @ F9 )
& ! [X4: A] :
( ( Q6 @ ( F4 @ X4 ) )
=> ( P4 @ X4 ) ) ) ) ) ) ).
% filtercomap_def
thf(fact_5865_inf__filter__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( filter @ A ) )
= ( ^ [F9: filter @ A,F10: filter @ A] :
( abs_filter @ A
@ ^ [P4: A > $o] :
? [Q6: A > $o,R6: A > $o] :
( ( eventually @ A @ Q6 @ F9 )
& ( eventually @ A @ R6 @ F10 )
& ! [X4: A] :
( ( ( Q6 @ X4 )
& ( R6 @ X4 ) )
=> ( P4 @ X4 ) ) ) ) ) ) ).
% inf_filter_def
thf(fact_5866_lex__take__index,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( lex @ A @ R2 ) )
=> ~ ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( take @ A @ I3 @ Xs2 )
= ( take @ A @ I3 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs2 @ I3 ) @ ( nth @ A @ Ys @ I3 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_5867_take__bit__horner__sum__bit__eq,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,Bs: list @ $o] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) )
= ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( take @ $o @ N @ Bs ) ) ) ) ).
% take_bit_horner_sum_bit_eq
thf(fact_5868_take__Suc__conv__app__nth,axiom,
! [A: $tType,I: nat,Xs2: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( take @ A @ ( suc @ I ) @ Xs2 )
= ( append @ A @ ( take @ A @ I @ Xs2 ) @ ( cons @ A @ ( nth @ A @ Xs2 @ I ) @ ( nil @ A ) ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_5869_nth__repl,axiom,
! [A: $tType,M2: nat,Xs2: list @ A,N: nat,X3: A] :
( ( ord_less @ nat @ M2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( M2 != N )
=> ( ( nth @ A @ ( append @ A @ ( take @ A @ N @ Xs2 ) @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( drop @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ Xs2 ) ) ) @ M2 )
= ( nth @ A @ Xs2 @ M2 ) ) ) ) ) ).
% nth_repl
thf(fact_5870_pos__n__replace,axiom,
! [A: $tType,N: nat,Xs2: list @ A,Y: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ ( append @ A @ ( take @ A @ N @ Xs2 ) @ ( append @ A @ ( cons @ A @ Y @ ( nil @ A ) ) @ ( drop @ A @ ( suc @ N ) @ Xs2 ) ) ) ) ) ) ).
% pos_n_replace
thf(fact_5871_upd__conv__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs2: list @ A,A3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ Xs2 @ I @ A3 )
= ( append @ A @ ( take @ A @ I @ Xs2 ) @ ( cons @ A @ A3 @ ( drop @ A @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_5872_drop__drop,axiom,
! [A: $tType,N: nat,M2: nat,Xs2: list @ A] :
( ( drop @ A @ N @ ( drop @ A @ M2 @ Xs2 ) )
= ( drop @ A @ ( plus_plus @ nat @ N @ M2 ) @ Xs2 ) ) ).
% drop_drop
thf(fact_5873_drop__Suc__Cons,axiom,
! [A: $tType,N: nat,X3: A,Xs2: list @ A] :
( ( drop @ A @ ( suc @ N ) @ ( cons @ A @ X3 @ Xs2 ) )
= ( drop @ A @ N @ Xs2 ) ) ).
% drop_Suc_Cons
thf(fact_5874_length__drop,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( drop @ A @ N @ Xs2 ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ).
% length_drop
thf(fact_5875_drop__all,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N )
=> ( ( drop @ A @ N @ Xs2 )
= ( nil @ A ) ) ) ).
% drop_all
thf(fact_5876_drop__eq__Nil,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( drop @ A @ N @ Xs2 )
= ( nil @ A ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ).
% drop_eq_Nil
thf(fact_5877_drop__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( nil @ A )
= ( drop @ A @ N @ Xs2 ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_5878_drop__append,axiom,
! [A: $tType,N: nat,Xs2: list @ A,Ys: list @ A] :
( ( drop @ A @ N @ ( append @ A @ Xs2 @ Ys ) )
= ( append @ A @ ( drop @ A @ N @ Xs2 ) @ ( drop @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_5879_drop__Cons__numeral,axiom,
! [A: $tType,V2: num,X3: A,Xs2: list @ A] :
( ( drop @ A @ ( numeral_numeral @ nat @ V2 ) @ ( cons @ A @ X3 @ Xs2 ) )
= ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) @ Xs2 ) ) ).
% drop_Cons_numeral
thf(fact_5880_nth__drop,axiom,
! [A: $tType,N: nat,Xs2: list @ A,I: nat] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( drop @ A @ N @ Xs2 ) @ I )
= ( nth @ A @ Xs2 @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_5881_take__drop,axiom,
! [A: $tType,N: nat,M2: nat,Xs2: list @ A] :
( ( take @ A @ N @ ( drop @ A @ M2 @ Xs2 ) )
= ( drop @ A @ M2 @ ( take @ A @ ( plus_plus @ nat @ N @ M2 ) @ Xs2 ) ) ) ).
% take_drop
thf(fact_5882_in__set__dropD,axiom,
! [A: $tType,X3: A,N: nat,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( drop @ A @ N @ Xs2 ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) ) ) ).
% in_set_dropD
thf(fact_5883_set__drop__subset,axiom,
! [A: $tType,N: nat,Xs2: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ N @ Xs2 ) ) @ ( set2 @ A @ Xs2 ) ) ).
% set_drop_subset
thf(fact_5884_drop__eq__nths,axiom,
! [A: $tType] :
( ( drop @ A )
= ( ^ [N3: nat,Xs: list @ A] : ( nths @ A @ Xs @ ( collect @ nat @ ( ord_less_eq @ nat @ N3 ) ) ) ) ) ).
% drop_eq_nths
thf(fact_5885_set__drop__subset__set__drop,axiom,
! [A: $tType,N: nat,M2: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ M2 @ Xs2 ) ) @ ( set2 @ A @ ( drop @ A @ N @ Xs2 ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_5886_drop__update__swap,axiom,
! [A: $tType,M2: nat,N: nat,Xs2: list @ A,X3: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( drop @ A @ M2 @ ( list_update @ A @ Xs2 @ N @ X3 ) )
= ( list_update @ A @ ( drop @ A @ M2 @ Xs2 ) @ ( minus_minus @ nat @ N @ M2 ) @ X3 ) ) ) ).
% drop_update_swap
thf(fact_5887_append__eq__conv__conj,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( append @ A @ Xs2 @ Ys )
= Zs2 )
= ( ( Xs2
= ( take @ A @ ( size_size @ ( list @ A ) @ Xs2 ) @ Zs2 ) )
& ( Ys
= ( drop @ A @ ( size_size @ ( list @ A ) @ Xs2 ) @ Zs2 ) ) ) ) ).
% append_eq_conv_conj
thf(fact_5888_take__add,axiom,
! [A: $tType,I: nat,J: nat,Xs2: list @ A] :
( ( take @ A @ ( plus_plus @ nat @ I @ J ) @ Xs2 )
= ( append @ A @ ( take @ A @ I @ Xs2 ) @ ( take @ A @ J @ ( drop @ A @ I @ Xs2 ) ) ) ) ).
% take_add
thf(fact_5889_nths__drop,axiom,
! [A: $tType,N: nat,Xs2: list @ A,I5: set @ nat] :
( ( nths @ A @ ( drop @ A @ N @ Xs2 ) @ I5 )
= ( nths @ A @ Xs2 @ ( image2 @ nat @ nat @ ( plus_plus @ nat @ N ) @ I5 ) ) ) ).
% nths_drop
thf(fact_5890_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_5891_Cons__nth__drop__Suc,axiom,
! [A: $tType,I: nat,Xs2: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( cons @ A @ ( nth @ A @ Xs2 @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs2 ) )
= ( drop @ A @ I @ Xs2 ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_5892_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_5893_id__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs2: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( Xs2
= ( append @ A @ ( take @ A @ I @ Xs2 ) @ ( cons @ A @ ( nth @ A @ Xs2 @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_5894_take__hd__drop,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( append @ A @ ( take @ A @ N @ Xs2 ) @ ( cons @ A @ ( hd @ A @ ( drop @ A @ N @ Xs2 ) ) @ ( nil @ A ) ) )
= ( take @ A @ ( suc @ N ) @ Xs2 ) ) ) ).
% take_hd_drop
thf(fact_5895_lexord__take__index__conv,axiom,
! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X3 ) @ ( size_size @ ( list @ A ) @ Y ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X3 ) @ Y )
= X3 ) )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X3 ) @ ( size_size @ ( list @ A ) @ Y ) ) )
& ( ( take @ A @ I4 @ X3 )
= ( take @ A @ I4 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X3 @ I4 ) @ ( nth @ A @ Y @ I4 ) ) @ R2 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_5896_rotate__drop__take,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N3: nat,Xs: list @ A] : ( append @ A @ ( drop @ A @ ( modulo_modulo @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) ) @ Xs ) @ ( take @ A @ ( modulo_modulo @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) ) @ Xs ) ) ) ) ).
% rotate_drop_take
thf(fact_5897_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C3 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% min.bounded_iff
thf(fact_5898_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_min @ A @ A3 @ B2 )
= B2 ) ) ) ).
% min.absorb2
thf(fact_5899_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_min @ A @ A3 @ B2 )
= A3 ) ) ) ).
% min.absorb1
thf(fact_5900_min__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X3: A] :
( ( ord_min @ A @ ( top_top @ A ) @ X3 )
= X3 ) ) ).
% min_top
thf(fact_5901_min__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X3: A] :
( ( ord_min @ A @ X3 @ ( top_top @ A ) )
= X3 ) ) ).
% min_top2
thf(fact_5902_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X3 )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_5903_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ord_min @ A @ X3 @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_5904_max__min__same_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X3: A] :
( ( ord_max @ A @ Y @ ( ord_min @ A @ X3 @ Y ) )
= Y ) ) ).
% max_min_same(4)
thf(fact_5905_max__min__same_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X3 @ Y ) @ Y )
= Y ) ) ).
% max_min_same(3)
thf(fact_5906_max__min__same_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X3 @ Y ) @ X3 )
= X3 ) ) ).
% max_min_same(2)
thf(fact_5907_max__min__same_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_max @ A @ X3 @ ( ord_min @ A @ X3 @ Y ) )
= X3 ) ) ).
% max_min_same(1)
thf(fact_5908_min__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_min @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( suc @ ( ord_min @ nat @ M2 @ N ) ) ) ).
% min_Suc_Suc
thf(fact_5909_min__0L,axiom,
! [N: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_5910_min__0R,axiom,
! [N: nat] :
( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_5911_set__rotate,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( set2 @ A @ ( rotate @ A @ N @ Xs2 ) )
= ( set2 @ A @ Xs2 ) ) ).
% set_rotate
thf(fact_5912_length__rotate,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate @ A @ N @ Xs2 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_rotate
thf(fact_5913_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_5914_min__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X3 ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(3)
thf(fact_5915_min__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X3 ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(4)
thf(fact_5916_min__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X3 ) )
= ( one_one @ A ) ) ) ).
% min_0_1(5)
thf(fact_5917_min__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X3 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% min_0_1(6)
thf(fact_5918_length__take,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( take @ A @ N @ Xs2 ) )
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ).
% length_take
thf(fact_5919_rotate__Suc,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( rotate @ A @ ( suc @ N ) @ Xs2 )
= ( rotate1 @ A @ ( rotate @ A @ N @ Xs2 ) ) ) ).
% rotate_Suc
thf(fact_5920_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_5921_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_5922_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_5923_rotate__length01,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) )
=> ( ( rotate @ A @ N @ Xs2 )
= Xs2 ) ) ).
% rotate_length01
thf(fact_5924_rotate__id,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
= ( zero_zero @ nat ) )
=> ( ( rotate @ A @ N @ Xs2 )
= Xs2 ) ) ).
% rotate_id
thf(fact_5925_min__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_min @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( suc @ N ) )
= ( suc @ ( ord_min @ nat @ ( pred_numeral @ K2 ) @ N ) ) ) ).
% min_numeral_Suc
thf(fact_5926_min__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_min @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K2 ) )
= ( suc @ ( ord_min @ nat @ N @ ( pred_numeral @ K2 ) ) ) ) ).
% min_Suc_numeral
thf(fact_5927_min__diff,axiom,
! [M2: nat,I: nat,N: nat] :
( ( ord_min @ nat @ ( minus_minus @ nat @ M2 @ I ) @ ( minus_minus @ nat @ N @ I ) )
= ( minus_minus @ nat @ ( ord_min @ nat @ M2 @ N ) @ I ) ) ).
% min_diff
thf(fact_5928_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 )
= ( ord_min @ A @ ( minus_minus @ A @ X3 @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ).
% min_diff_distrib_left
thf(fact_5929_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( ord_min @ A @ X3 @ Y )
= Y ) ) ) ).
% min_absorb2
thf(fact_5930_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_min @ A @ X3 @ Y )
= X3 ) ) ) ).
% min_absorb1
thf(fact_5931_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A8: A,B8: A] : ( if @ A @ ( ord_less_eq @ A @ A8 @ B8 ) @ A8 @ B8 ) ) ) ) ).
% min_def
thf(fact_5932_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X3 @ Z2 )
| ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% min_le_iff_disj
thf(fact_5933_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% min.coboundedI2
thf(fact_5934_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% min.coboundedI1
thf(fact_5935_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B8: A,A8: A] :
( ( ord_min @ A @ A8 @ B8 )
= B8 ) ) ) ) ).
% min.absorb_iff2
thf(fact_5936_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] :
( ( ord_min @ A @ A8 @ B8 )
= A8 ) ) ) ) ).
% min.absorb_iff1
thf(fact_5937_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 ) ) ).
% min.cobounded2
thf(fact_5938_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ A3 ) ) ).
% min.cobounded1
thf(fact_5939_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] :
( A8
= ( ord_min @ A @ A8 @ B8 ) ) ) ) ) ).
% min.order_iff
thf(fact_5940_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C3 ) ) ) ) ) ).
% min.boundedI
thf(fact_5941_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C3 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% min.boundedE
thf(fact_5942_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( ord_min @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% min.orderI
thf(fact_5943_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3
= ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% min.orderE
thf(fact_5944_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C3: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ ( ord_min @ A @ C3 @ D3 ) ) ) ) ) ).
% min.mono
thf(fact_5945_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A8: A,B8: A] : ( if @ A @ ( ord_less_eq @ A @ A8 @ B8 ) @ A8 @ B8 ) ) ) ) ).
% min_def_raw
thf(fact_5946_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X3: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X3 @ Y ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_max_eq_min
thf(fact_5947_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X3: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X3 @ Y ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_min_eq_max
thf(fact_5948_of__nat__min,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X3 @ Y ) )
= ( ord_min @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_min
thf(fact_5949_inf__nat__def,axiom,
( ( inf_inf @ nat )
= ( ord_min @ nat ) ) ).
% inf_nat_def
thf(fact_5950_nat__mult__min__right,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( times_times @ nat @ M2 @ ( ord_min @ nat @ N @ Q3 ) )
= ( ord_min @ nat @ ( times_times @ nat @ M2 @ N ) @ ( times_times @ nat @ M2 @ Q3 ) ) ) ).
% nat_mult_min_right
thf(fact_5951_nat__mult__min__left,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( times_times @ nat @ ( ord_min @ nat @ M2 @ N ) @ Q3 )
= ( ord_min @ nat @ ( times_times @ nat @ M2 @ Q3 ) @ ( times_times @ nat @ N @ Q3 ) ) ) ).
% nat_mult_min_left
thf(fact_5952_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 )
= ( ord_min @ A @ ( plus_plus @ A @ X3 @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ).
% min_add_distrib_left
thf(fact_5953_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( plus_plus @ A @ X3 @ ( ord_min @ A @ Y @ Z2 ) )
= ( ord_min @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( plus_plus @ A @ X3 @ Z2 ) ) ) ) ).
% min_add_distrib_right
thf(fact_5954_rotate__rotate,axiom,
! [A: $tType,M2: nat,N: nat,Xs2: list @ A] :
( ( rotate @ A @ M2 @ ( rotate @ A @ N @ Xs2 ) )
= ( rotate @ A @ ( plus_plus @ nat @ M2 @ N ) @ Xs2 ) ) ).
% rotate_rotate
thf(fact_5955_rotate__add,axiom,
! [A: $tType,M2: nat,N: nat] :
( ( rotate @ A @ ( plus_plus @ nat @ M2 @ N ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( rotate @ A @ M2 ) @ ( rotate @ A @ N ) ) ) ).
% rotate_add
thf(fact_5956_list_Oset__sel_I1_J,axiom,
! [A: $tType,A3: list @ A] :
( ( A3
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ A3 ) @ ( set2 @ A @ A3 ) ) ) ).
% list.set_sel(1)
thf(fact_5957_hd__in__set,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ Xs2 ) @ ( set2 @ A @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_5958_hd__rotate__conv__nth,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( hd @ A @ ( rotate @ A @ N @ Xs2 ) )
= ( nth @ A @ Xs2 @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_5959_rotate__append,axiom,
! [A: $tType,L: list @ A,Q3: list @ A] :
( ( rotate @ A @ ( size_size @ ( list @ A ) @ L ) @ ( append @ A @ L @ Q3 ) )
= ( append @ A @ Q3 @ L ) ) ).
% rotate_append
thf(fact_5960_rotate__conv__mod,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N3: nat,Xs: list @ A] : ( rotate @ A @ ( modulo_modulo @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) ) @ Xs ) ) ) ).
% rotate_conv_mod
thf(fact_5961_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P2: A,X3: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( times_times @ A @ P2 @ ( ord_max @ A @ X3 @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P2 @ X3 ) @ ( times_times @ A @ P2 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( times_times @ A @ P2 @ ( ord_max @ A @ X3 @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P2 @ X3 ) @ ( times_times @ A @ P2 @ Y ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_5962_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P2: A,X3: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( times_times @ A @ P2 @ ( ord_min @ A @ X3 @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P2 @ X3 ) @ ( times_times @ A @ P2 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( times_times @ A @ P2 @ ( ord_min @ A @ X3 @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P2 @ X3 ) @ ( times_times @ A @ P2 @ Y ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_5963_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P2: A,X3: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( times_times @ A @ ( ord_max @ A @ X3 @ Y ) @ P2 )
= ( ord_max @ A @ ( times_times @ A @ X3 @ P2 ) @ ( times_times @ A @ Y @ P2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( times_times @ A @ ( ord_max @ A @ X3 @ Y ) @ P2 )
= ( ord_min @ A @ ( times_times @ A @ X3 @ P2 ) @ ( times_times @ A @ Y @ P2 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_5964_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P2: A,X3: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( times_times @ A @ ( ord_min @ A @ X3 @ Y ) @ P2 )
= ( ord_min @ A @ ( times_times @ A @ X3 @ P2 ) @ ( times_times @ A @ Y @ P2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( times_times @ A @ ( ord_min @ A @ X3 @ Y ) @ P2 )
= ( ord_max @ A @ ( times_times @ A @ X3 @ P2 ) @ ( times_times @ A @ Y @ P2 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_5965_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P2: A,X3: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X3 @ Y ) @ P2 )
= ( ord_max @ A @ ( divide_divide @ A @ X3 @ P2 ) @ ( divide_divide @ A @ Y @ P2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X3 @ Y ) @ P2 )
= ( ord_min @ A @ ( divide_divide @ A @ X3 @ P2 ) @ ( divide_divide @ A @ Y @ P2 ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_5966_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P2: A,X3: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X3 @ Y ) @ P2 )
= ( ord_min @ A @ ( divide_divide @ A @ X3 @ P2 ) @ ( divide_divide @ A @ Y @ P2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X3 @ Y ) @ P2 )
= ( ord_max @ A @ ( divide_divide @ A @ X3 @ P2 ) @ ( divide_divide @ A @ Y @ P2 ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_5967_min__Suc1,axiom,
! [N: nat,M2: nat] :
( ( ord_min @ nat @ ( suc @ N ) @ M2 )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M6: nat] : ( suc @ ( ord_min @ nat @ N @ M6 ) )
@ M2 ) ) ).
% min_Suc1
thf(fact_5968_min__Suc2,axiom,
! [M2: nat,N: nat] :
( ( ord_min @ nat @ M2 @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M6: nat] : ( suc @ ( ord_min @ nat @ M6 @ N ) )
@ M2 ) ) ).
% min_Suc2
thf(fact_5969_Inf__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S3: set @ A,X3: A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( ( S3
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ X3 @ S3 ) )
= X3 ) )
& ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ X3 @ S3 ) )
= ( ord_min @ A @ X3 @ ( complete_Inf_Inf @ A @ S3 ) ) ) ) ) ) ) ).
% Inf_insert_finite
thf(fact_5970_hd__drop__conv__nth,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( hd @ A @ ( drop @ A @ N @ Xs2 ) )
= ( nth @ A @ Xs2 @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_5971_mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M2 @ N ) ) ) ) ) ).
% mod_exp_eq
thf(fact_5972_nth__rotate,axiom,
! [A: $tType,N: nat,Xs2: list @ A,M2: nat] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( rotate @ A @ M2 @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ).
% nth_rotate
thf(fact_5973_mask__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat,M2: nat] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M2 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% mask_mod_exp
thf(fact_5974_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( ord_min @ A @ X4 ) @ Y3 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Min.eq_fold'
thf(fact_5975_set__zip,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [I4: nat] :
( ( Uu3
= ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ I4 ) @ ( nth @ B @ Ys @ I4 ) ) )
& ( ord_less @ nat @ I4 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% set_zip
thf(fact_5976_dual__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( max @ A
@ ^ [X4: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X4 ) )
= ( ord_min @ A ) ) ) ).
% dual_max
thf(fact_5977_Min__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A] :
( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% Min_singleton
thf(fact_5978_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A6 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ X3 @ X4 ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_5979_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A6 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less @ A @ X3 @ X4 ) ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_5980_zip__replicate,axiom,
! [A: $tType,B: $tType,I: nat,X3: A,J: nat,Y: B] :
( ( zip @ A @ B @ ( replicate @ A @ I @ X3 ) @ ( replicate @ B @ J @ Y ) )
= ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I @ J ) @ ( product_Pair @ A @ B @ X3 @ Y ) ) ) ).
% zip_replicate
thf(fact_5981_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X3: A,Xs2: list @ A,Y: B,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y @ Ys ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_5982_Min__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ B,C3: A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [Uu3: B] : C3
@ A6 ) )
= C3 ) ) ) ) ).
% Min_const
thf(fact_5983_zip__append,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Us: list @ B,Ys: list @ A,Vs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Us ) )
=> ( ( zip @ A @ B @ ( append @ A @ Xs2 @ Ys ) @ ( append @ B @ Us @ Vs ) )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Us ) @ ( zip @ A @ B @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_5984_length__zip,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( zip @ A @ B @ Xs2 @ Ys ) )
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% length_zip
thf(fact_5985_Min__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% Min_insert
thf(fact_5986_minus__Min__eq__Max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798350308766er_Min @ A @ S3 ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ S3 ) ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_5987_minus__Max__eq__Min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798349783984er_Max @ A @ S3 ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ S3 ) ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_5988_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs2: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) @ I )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ I ) @ ( nth @ B @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_5989_Min__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ A6 ) ) ) ) ).
% Min_in
thf(fact_5990_Min__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ X3 ) ) ) ) ).
% Min_le
thf(fact_5991_Min__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ A6 )
=> ( ord_less_eq @ A @ X3 @ Y4 ) )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( lattic643756798350308766er_Min @ A @ A6 )
= X3 ) ) ) ) ) ).
% Min_eqI
thf(fact_5992_Min_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ A3 ) ) ) ) ).
% Min.coboundedI
thf(fact_5993_zip__update,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,I: nat,X3: A,Ys: list @ B,Y: B] :
( ( zip @ A @ B @ ( list_update @ A @ Xs2 @ I @ X3 ) @ ( list_update @ B @ Ys @ I @ Y ) )
= ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) @ I @ ( product_Pair @ A @ B @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_5994_ord_Omax__def,axiom,
! [A: $tType] :
( ( max @ A )
= ( ^ [Less_eq: A > A > $o,A8: A,B8: A] : ( if @ A @ ( Less_eq @ A8 @ B8 ) @ B8 @ A8 ) ) ) ).
% ord.max_def
thf(fact_5995_ord_Omax_Ocong,axiom,
! [A: $tType] :
( ( max @ A )
= ( max @ A ) ) ).
% ord.max.cong
thf(fact_5996_hd__zip,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ( hd @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) )
= ( product_Pair @ A @ B @ ( hd @ A @ Xs2 ) @ ( hd @ B @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_5997_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X3: A,Y: B,Xs2: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) )
=> ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_5998_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X3: A,Y: B,Xs2: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) ) ) ).
% set_zip_leftD
thf(fact_5999_in__set__zipE,axiom,
! [A: $tType,B: $tType,X3: A,Y: B,Xs2: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) )
=> ~ ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ~ ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_6000_zip__same,axiom,
! [A: $tType,A3: A,B2: A,Xs2: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs2 @ Xs2 ) ) )
= ( ( member @ A @ A3 @ ( set2 @ A @ Xs2 ) )
& ( A3 = B2 ) ) ) ).
% zip_same
thf(fact_6001_zip__obtain__same__length,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,P: ( list @ ( product_prod @ A @ B ) ) > $o] :
( ! [Zs: list @ A,Ws2: list @ B,N2: nat] :
( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ B ) @ Ws2 ) )
=> ( ( N2
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( Zs
= ( take @ A @ N2 @ Xs2 ) )
=> ( ( Ws2
= ( take @ B @ N2 @ Ys ) )
=> ( P @ ( zip @ A @ B @ Zs @ Ws2 ) ) ) ) ) )
=> ( P @ ( zip @ A @ B @ Xs2 @ Ys ) ) ) ).
% zip_obtain_same_length
thf(fact_6002_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ( ord_less_eq @ A @ X3 @ A5 ) )
=> ( ord_less_eq @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_6003_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A6 ) )
=> ! [A18: A] :
( ( member @ A @ A18 @ A6 )
=> ( ord_less_eq @ A @ X3 @ A18 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_6004_eq__Min__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,M2: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M2
= ( lattic643756798350308766er_Min @ A @ A6 ) )
= ( ( member @ A @ M2 @ A6 )
& ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ M2 @ X4 ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_6005_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ X3 )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ord_less_eq @ A @ X4 @ X3 ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_6006_Min__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,M2: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798350308766er_Min @ A @ A6 )
= M2 )
= ( ( member @ A @ M2 @ A6 )
& ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ M2 @ X4 ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_6007_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ X3 )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ord_less @ A @ X4 @ X3 ) ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_6008_Min__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ A6 )
=> ( ord_less_eq @ A @ A3 @ B4 ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ A3 @ A6 ) )
= A3 ) ) ) ) ).
% Min_insert2
thf(fact_6009_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,Xy2: product_prod @ A @ B,Xys2: list @ ( product_prod @ A @ B )] :
( ( ( zip @ A @ B @ Xs2 @ Ys )
= ( cons @ ( product_prod @ A @ B ) @ Xy2 @ Xys2 ) )
=> ~ ! [X5: A,Xs5: list @ A] :
( ( Xs2
= ( cons @ A @ X5 @ Xs5 ) )
=> ! [Y4: B,Ys6: list @ B] :
( ( Ys
= ( cons @ B @ Y4 @ Ys6 ) )
=> ( ( Xy2
= ( product_Pair @ A @ B @ X5 @ Y4 ) )
=> ( Xys2
!= ( zip @ A @ B @ Xs5 @ Ys6 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_6010_cInf__eq__Min,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= ( lattic643756798350308766er_Min @ A @ X6 ) ) ) ) ) ).
% cInf_eq_Min
thf(fact_6011_Min__Inf,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A6 )
= ( complete_Inf_Inf @ A @ A6 ) ) ) ) ) ).
% Min_Inf
thf(fact_6012_Min_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( lattic643756798350308766er_Min @ A @ A6 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Min.infinite
thf(fact_6013_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,Y: B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ B @ Y @ ( set2 @ B @ Ys ) )
=> ~ ! [X5: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_6014_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,X3: A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ~ ! [Y4: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_6015_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B5 ) @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_6016_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M7: set @ A,N5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M7 @ N5 )
=> ( ( M7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N5 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N5 ) @ ( lattic643756798350308766er_Min @ A @ M7 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_6017_hom__Min__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H: A > A,N5: set @ A] :
( ! [X5: A,Y4: A] :
( ( H @ ( ord_min @ A @ X5 @ Y4 ) )
= ( ord_min @ A @ ( H @ X5 ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite2 @ A @ N5 )
=> ( ( N5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic643756798350308766er_Min @ A @ N5 ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ H @ N5 ) ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_6018_Min_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B5 ) @ ( lattic643756798350308766er_Min @ A @ A6 ) )
= ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% Min.subset
thf(fact_6019_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X3 @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_6020_Min_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y4: A] : ( member @ A @ ( ord_min @ A @ X5 @ Y4 ) @ ( insert2 @ A @ X5 @ ( insert2 @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ A6 ) ) ) ) ) ).
% Min.closed
thf(fact_6021_Min_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ ( lattic643756798350308766er_Min @ A @ B5 ) ) ) ) ) ) ) ) ).
% Min.union
thf(fact_6022_Min_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( finite_fold @ A @ A @ ( ord_min @ A ) @ X3 @ A6 ) ) ) ) ).
% Min.eq_fold
thf(fact_6023_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S3: set @ B,F3: B > A,K2: A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F3 @ X4 ) @ K2 )
@ S3 ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image2 @ B @ A @ F3 @ S3 ) ) @ K2 ) ) ) ) ) ).
% Min_add_commute
thf(fact_6024_list__eq__iff__zip__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z: list @ A] : Y5 = Z )
= ( ^ [Xs: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs @ Ys3 ) ) )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y5: A,Z: A] : Y5 = Z
@ X4 ) ) ) ) ) ).
% list_eq_iff_zip_eq
thf(fact_6025_concat__eq__concat__iff,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ! [X5: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X5 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y3: list @ A,Z4: list @ A] :
( ( size_size @ ( list @ A ) @ Y3 )
= ( size_size @ ( list @ A ) @ Z4 ) )
@ X5 ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs2 )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys ) )
=> ( ( ( concat @ A @ Xs2 )
= ( concat @ A @ Ys ) )
= ( Xs2 = Ys ) ) ) ) ).
% concat_eq_concat_iff
thf(fact_6026_concat__injective,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ( ( concat @ A @ Xs2 )
= ( concat @ A @ Ys ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs2 )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys ) )
=> ( ! [X5: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X5 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y3: list @ A,Z4: list @ A] :
( ( size_size @ ( list @ A ) @ Y3 )
= ( size_size @ ( list @ A ) @ Z4 ) )
@ X5 ) )
=> ( Xs2 = Ys ) ) ) ) ).
% concat_injective
thf(fact_6027_Min_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X3 @ A6 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_6028_Min_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A6 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A6 )
= ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_6029_zip__append2,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,Zs2: list @ B] :
( ( zip @ A @ B @ Xs2 @ ( append @ B @ Ys @ Zs2 ) )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs2 ) @ Ys ) @ ( zip @ A @ B @ ( drop @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs2 ) @ Zs2 ) ) ) ).
% zip_append2
thf(fact_6030_zip__append1,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ A,Zs2: list @ B] :
( ( zip @ A @ B @ ( append @ A @ Xs2 @ Ys ) @ Zs2 )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ ( take @ B @ ( size_size @ ( list @ A ) @ Xs2 ) @ Zs2 ) ) @ ( zip @ A @ B @ Ys @ ( drop @ B @ ( size_size @ ( list @ A ) @ Xs2 ) @ Zs2 ) ) ) ) ).
% zip_append1
thf(fact_6031_in__set__zip,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,Xs2: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ P2 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) )
= ( ? [N3: nat] :
( ( ( nth @ A @ Xs2 @ N3 )
= ( product_fst @ A @ B @ P2 ) )
& ( ( nth @ B @ Ys @ N3 )
= ( product_snd @ A @ B @ P2 ) )
& ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ord_less @ nat @ N3 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ).
% in_set_zip
thf(fact_6032_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( linord4507533701916653071of_set @ A @ A6 )
= ( cons @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_6033_card__Min__le__sum,axiom,
! [A: $tType,A6: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( finite_card @ A @ A6 ) @ ( lattic643756798350308766er_Min @ nat @ ( image2 @ A @ nat @ F3 @ A6 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A6 ) ) ) ).
% card_Min_le_sum
thf(fact_6034_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [X4: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X4 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) )
=> ( product_case_prod @ A @ B @ $o
@ ^ [Y3: A,Z4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Z4 ) @ R2 )
@ X4 ) ) ) ) ).
% listrel_iff_zip
thf(fact_6035_dual__Max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Max @ A
@ ^ [X4: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X4 ) )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% dual_Max
thf(fact_6036_listrel__rtrancl__refl,axiom,
! [A: $tType,Xs2: list @ A,R2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Xs2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) ).
% listrel_rtrancl_refl
thf(fact_6037_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ ( nil @ B ) ) @ ( listrel @ A @ B @ R2 ) )
=> ( Xs2
= ( nil @ A ) ) ) ).
% listrel_Nil2
thf(fact_6038_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs2: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs2 ) @ ( listrel @ A @ B @ R2 ) )
=> ( Xs2
= ( nil @ B ) ) ) ).
% listrel_Nil1
thf(fact_6039_listrel_ONil,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R2 ) ) ).
% listrel.Nil
thf(fact_6040_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys ) @ ( listrel @ A @ B @ R2 ) )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_6041_listrel__rtrancl__trans,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),Zs2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Zs2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_6042_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Y: B,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R2 ) )
=> ~ ! [X5: A,Xs3: list @ A] :
( ( Xs2
= ( cons @ A @ X5 @ Xs3 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_6043_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys: list @ A,Xs2: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y @ Ys ) @ Xs2 ) @ ( listrel @ A @ B @ R2 ) )
=> ~ ! [Y4: B,Ys4: list @ B] :
( ( Xs2
= ( cons @ B @ Y4 @ Ys4 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y4 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys @ Ys4 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_6044_listrel_OCons,axiom,
! [B: $tType,A: $tType,X3: A,Y: B,R2: set @ ( product_prod @ A @ B ),Xs2: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys ) @ ( listrel @ A @ B @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_6045_listrel__reflcl__if__listrel1,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% listrel_reflcl_if_listrel1
thf(fact_6046_rtrancl__listrel1__if__listrel,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel @ A @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ).
% rtrancl_listrel1_if_listrel
thf(fact_6047_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R2 ) )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X5: A,Y4: B,Xs3: list @ A] :
( ( A1
= ( cons @ A @ X5 @ Xs3 ) )
=> ! [Ys4: list @ B] :
( ( A22
= ( cons @ B @ Y4 @ Ys4 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys4 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_6048_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( A1
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X4: A,Y3: B,Xs: list @ A,Ys3: list @ B] :
( ( A1
= ( cons @ A @ X4 @ Xs ) )
& ( A22
= ( cons @ B @ Y3 @ Ys3 ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys3 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_6049_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ N3 ) @ ( nth @ B @ Ys @ N3 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_6050_f__arg__min__list__f,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [Xs2: list @ A,F3: A > B] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( F3 @ ( arg_min_list @ A @ B @ F3 @ Xs2 ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F3 @ ( set2 @ A @ Xs2 ) ) ) ) ) ) ).
% f_arg_min_list_f
thf(fact_6051_map__upds__fold__map__upd,axiom,
! [B: $tType,A: $tType] :
( ( map_upds @ A @ B )
= ( ^ [M5: A > ( option @ B ),Ks: list @ A,Vs2: list @ B] :
( foldl @ ( A > ( option @ B ) ) @ ( product_prod @ A @ B )
@ ^ [N3: A > ( option @ B )] :
( product_case_prod @ A @ B @ ( A > ( option @ B ) )
@ ^ [K3: A,V5: B] : ( fun_upd @ A @ ( option @ B ) @ N3 @ K3 @ ( some @ B @ V5 ) ) )
@ M5
@ ( zip @ A @ B @ Ks @ Vs2 ) ) ) ) ).
% map_upds_fold_map_upd
thf(fact_6052_foldl__cong,axiom,
! [A: $tType,B: $tType,A3: A,B2: A,L: list @ B,K2: list @ B,F3: A > B > A,G3: A > B > A] :
( ( A3 = B2 )
=> ( ( L = K2 )
=> ( ! [A5: A,X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ L ) )
=> ( ( F3 @ A5 @ X5 )
= ( G3 @ A5 @ X5 ) ) )
=> ( ( foldl @ A @ B @ F3 @ A3 @ L )
= ( foldl @ A @ B @ G3 @ B2 @ K2 ) ) ) ) ) ).
% foldl_cong
thf(fact_6053_arg__min__list__in,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [Xs2: list @ A,F3: A > B] :
( ( Xs2
!= ( nil @ A ) )
=> ( member @ A @ ( arg_min_list @ A @ B @ F3 @ Xs2 ) @ ( set2 @ A @ Xs2 ) ) ) ) ).
% arg_min_list_in
thf(fact_6054_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,X3: A,Y: A,Zs2: list @ A] :
( ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Zs2 ) ) )
= ( if @ A @ ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ Y @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ Y @ Zs2 ) ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_6055_min__list__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( min_list @ A @ Xs2 )
= ( lattic643756798350308766er_Min @ A @ ( set2 @ A @ Xs2 ) ) ) ) ) ).
% min_list_Min
thf(fact_6056_listrel__def,axiom,
! [B: $tType,A: $tType] :
( ( listrel @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ B ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ B ) @ $o
@ ( listrelp @ A @ B
@ ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R5 ) ) ) ) ) ) ).
% listrel_def
thf(fact_6057_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( listrelp @ A @ B
@ ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R2 ) )
= ( ^ [X4: list @ A,Y3: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X4 @ Y3 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_6058_dual__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( min @ A
@ ^ [X4: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X4 ) )
= ( ord_max @ A ) ) ) ).
% dual_min
thf(fact_6059_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,I: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( distinct @ A @ Xs2 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys ) @ ( nth @ A @ Xs2 @ I ) )
= ( some @ B @ ( nth @ B @ Ys @ I ) ) ) ) ) ) ).
% map_of_zip_nth
thf(fact_6060_map__of__eq__empty__iff,axiom,
! [B: $tType,A: $tType,Xys2: list @ ( product_prod @ A @ B )] :
( ( ( map_of @ A @ B @ Xys2 )
= ( ^ [X4: A] : ( none @ B ) ) )
= ( Xys2
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% map_of_eq_empty_iff
thf(fact_6061_empty__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys2: list @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A] : ( none @ B ) )
= ( map_of @ A @ B @ Xys2 ) )
= ( Xys2
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% empty_eq_map_of_iff
thf(fact_6062_map__of__zip__is__None,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B,X3: A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys ) @ X3 )
= ( none @ B ) )
= ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_6063_dom__map__of__zip,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( dom @ A @ B @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys ) ) )
= ( set2 @ A @ Xs2 ) ) ) ).
% dom_map_of_zip
thf(fact_6064_map__of__Cons__code_I1_J,axiom,
! [B: $tType,A: $tType,K2: B] :
( ( map_of @ B @ A @ ( nil @ ( product_prod @ B @ A ) ) @ K2 )
= ( none @ A ) ) ).
% map_of_Cons_code(1)
thf(fact_6065_map__of_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( map_of @ A @ B @ ( nil @ ( product_prod @ A @ B ) ) )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% map_of.simps(1)
thf(fact_6066_ord_Omin__def,axiom,
! [A: $tType] :
( ( min @ A )
= ( ^ [Less_eq: A > A > $o,A8: A,B8: A] : ( if @ A @ ( Less_eq @ A8 @ B8 ) @ A8 @ B8 ) ) ) ).
% ord.min_def
thf(fact_6067_ord_Omin_Ocong,axiom,
! [A: $tType] :
( ( min @ A )
= ( min @ A ) ) ).
% ord.min.cong
thf(fact_6068_map__of__SomeD,axiom,
! [A: $tType,B: $tType,Xs2: list @ ( product_prod @ B @ A ),K2: B,Y: A] :
( ( ( map_of @ B @ A @ Xs2 @ K2 )
= ( some @ A @ Y ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K2 @ Y ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs2 ) ) ) ).
% map_of_SomeD
thf(fact_6069_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K2: A,X3: B,L: list @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ X3 ) @ ( set2 @ ( product_prod @ A @ B ) @ L ) )
=> ? [X5: B] :
( ( map_of @ A @ B @ L @ K2 )
= ( some @ B @ X5 ) ) ) ).
% weak_map_of_SomeI
thf(fact_6070_map__of__zip__inject,axiom,
! [B: $tType,A: $tType,Ys: list @ A,Xs2: list @ B,Zs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys )
= ( size_size @ ( list @ B ) @ Xs2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs2 )
= ( size_size @ ( list @ B ) @ Xs2 ) )
=> ( ( distinct @ B @ Xs2 )
=> ( ( ( map_of @ B @ A @ ( zip @ B @ A @ Xs2 @ Ys ) )
= ( map_of @ B @ A @ ( zip @ B @ A @ Xs2 @ Zs2 ) ) )
=> ( Ys = Zs2 ) ) ) ) ) ).
% map_of_zip_inject
thf(fact_6071_map__of__Cons__code_I2_J,axiom,
! [C: $tType,B: $tType,L: B,K2: B,V2: C,Ps: list @ ( product_prod @ B @ C )] :
( ( ( L = K2 )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V2 ) @ Ps ) @ K2 )
= ( some @ C @ V2 ) ) )
& ( ( L != K2 )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V2 ) @ Ps ) @ K2 )
= ( map_of @ B @ C @ Ps @ K2 ) ) ) ) ).
% map_of_Cons_code(2)
thf(fact_6072_map__of__zip__is__Some,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,X3: A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
= ( ? [Y3: B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys ) @ X3 )
= ( some @ B @ Y3 ) ) ) ) ) ).
% map_of_zip_is_Some
thf(fact_6073_map__of__eq__None__iff,axiom,
! [A: $tType,B: $tType,Xys2: list @ ( product_prod @ B @ A ),X3: B] :
( ( ( map_of @ B @ A @ Xys2 @ X3 )
= ( none @ A ) )
= ( ~ ( member @ B @ X3 @ ( image2 @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ ( set2 @ ( product_prod @ B @ A ) @ Xys2 ) ) ) ) ) ).
% map_of_eq_None_iff
thf(fact_6074_map__of__zip__upd,axiom,
! [A: $tType,B: $tType,Ys: list @ B,Xs2: list @ A,Zs2: list @ B,X3: A,Y: B,Z2: B] :
( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Zs2 )
= ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys ) ) @ X3 @ ( some @ B @ Y ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Zs2 ) ) @ X3 @ ( some @ B @ Z2 ) ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys ) )
= ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Zs2 ) ) ) ) ) ) ) ).
% map_of_zip_upd
thf(fact_6075_ran__map__of__zip,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( distinct @ A @ Xs2 )
=> ( ( ran @ A @ B @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys ) ) )
= ( set2 @ B @ Ys ) ) ) ) ).
% ran_map_of_zip
thf(fact_6076_map__of_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,Ps: list @ ( product_prod @ A @ B )] :
( ( map_of @ A @ B @ ( cons @ ( product_prod @ A @ B ) @ P2 @ Ps ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Ps ) @ ( product_fst @ A @ B @ P2 ) @ ( some @ B @ ( product_snd @ A @ B @ P2 ) ) ) ) ).
% map_of.simps(2)
thf(fact_6077_map__of__eq__Some__iff,axiom,
! [B: $tType,A: $tType,Xys2: list @ ( product_prod @ A @ B ),X3: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys2 ) )
=> ( ( ( map_of @ A @ B @ Xys2 @ X3 )
= ( some @ B @ Y ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys2 ) ) ) ) ).
% map_of_eq_Some_iff
thf(fact_6078_Some__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys2: list @ ( product_prod @ A @ B ),Y: B,X3: A] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys2 ) )
=> ( ( ( some @ B @ Y )
= ( map_of @ A @ B @ Xys2 @ X3 ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys2 ) ) ) ) ).
% Some_eq_map_of_iff
thf(fact_6079_map__eq__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs2: list @ B,G3: B > A] :
( ( ( map @ B @ A @ F3 @ Xs2 )
= ( map @ B @ A @ G3 @ Xs2 ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Xs2 ) )
=> ( ( F3 @ X4 )
= ( G3 @ X4 ) ) ) ) ) ).
% map_eq_conv
thf(fact_6080_length__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs2: list @ B] :
( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F3 @ Xs2 ) )
= ( size_size @ ( list @ B ) @ Xs2 ) ) ).
% length_map
thf(fact_6081_list_Oset__map,axiom,
! [B: $tType,A: $tType,F3: A > B,V2: list @ A] :
( ( set2 @ B @ ( map @ A @ B @ F3 @ V2 ) )
= ( image2 @ A @ B @ F3 @ ( set2 @ A @ V2 ) ) ) ).
% list.set_map
thf(fact_6082_map__fun__upd,axiom,
! [B: $tType,A: $tType,Y: A,Xs2: list @ A,F3: A > B,V2: B] :
( ~ ( member @ A @ Y @ ( set2 @ A @ Xs2 ) )
=> ( ( map @ A @ B @ ( fun_upd @ A @ B @ F3 @ Y @ V2 ) @ Xs2 )
= ( map @ A @ B @ F3 @ Xs2 ) ) ) ).
% map_fun_upd
thf(fact_6083_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs2: list @ A,F3: A > B] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ B @ ( map @ A @ B @ F3 @ Xs2 ) @ N )
= ( F3 @ ( nth @ A @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_6084_map__fst__zip,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) )
= Xs2 ) ) ).
% map_fst_zip
thf(fact_6085_map__snd__zip,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_6086_map__of__is__SomeI,axiom,
! [A: $tType,B: $tType,Xys2: list @ ( product_prod @ A @ B ),X3: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys2 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys2 ) )
=> ( ( map_of @ A @ B @ Xys2 @ X3 )
= ( some @ B @ Y ) ) ) ) ).
% map_of_is_SomeI
thf(fact_6087_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: ( product_prod @ B @ C ) > A,Xs2: list @ B,G3: D > C,Ys: list @ D] :
( ( map @ ( product_prod @ B @ C ) @ A @ F3 @ ( zip @ B @ C @ Xs2 @ ( map @ D @ C @ G3 @ Ys ) ) )
= ( map @ ( product_prod @ B @ D ) @ A
@ ( product_case_prod @ B @ D @ A
@ ^ [X4: B,Y3: D] : ( F3 @ ( product_Pair @ B @ C @ X4 @ ( G3 @ Y3 ) ) ) )
@ ( zip @ B @ D @ Xs2 @ Ys ) ) ) ).
% map_zip_map2
thf(fact_6088_zip__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,F3: C > A,Xs2: list @ C,G3: D > B,Ys: list @ D] :
( ( zip @ A @ B @ ( map @ C @ A @ F3 @ Xs2 ) @ ( map @ D @ B @ G3 @ Ys ) )
= ( map @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X4: C,Y3: D] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ Y3 ) ) )
@ ( zip @ C @ D @ Xs2 @ Ys ) ) ) ).
% zip_map_map
thf(fact_6089_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F3: ( product_prod @ B @ C ) > A,G3: D > B,Xs2: list @ D,Ys: list @ C] :
( ( map @ ( product_prod @ B @ C ) @ A @ F3 @ ( zip @ B @ C @ ( map @ D @ B @ G3 @ Xs2 ) @ Ys ) )
= ( map @ ( product_prod @ D @ C ) @ A
@ ( product_case_prod @ D @ C @ A
@ ^ [X4: D,Y3: C] : ( F3 @ ( product_Pair @ B @ C @ ( G3 @ X4 ) @ Y3 ) ) )
@ ( zip @ D @ C @ Xs2 @ Ys ) ) ) ).
% map_zip_map
thf(fact_6090_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs2: list @ A,F3: C > B,Ys: list @ C] :
( ( zip @ A @ B @ Xs2 @ ( map @ C @ B @ F3 @ Ys ) )
= ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [X4: A,Y3: C] : ( product_Pair @ A @ B @ X4 @ ( F3 @ Y3 ) ) )
@ ( zip @ A @ C @ Xs2 @ Ys ) ) ) ).
% zip_map2
thf(fact_6091_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,Xs2: list @ C,Ys: list @ B] :
( ( zip @ A @ B @ ( map @ C @ A @ F3 @ Xs2 ) @ Ys )
= ( map @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ B @ ( product_prod @ A @ B )
@ ^ [X4: C] : ( product_Pair @ A @ B @ ( F3 @ X4 ) ) )
@ ( zip @ C @ B @ Xs2 @ Ys ) ) ) ).
% zip_map1
thf(fact_6092_enumerate__Suc__eq,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( enumerate @ A @ ( suc @ N ) @ Xs2 )
= ( map @ ( product_prod @ nat @ A ) @ ( product_prod @ nat @ A ) @ ( product_apfst @ nat @ nat @ A @ suc ) @ ( enumerate @ A @ N @ Xs2 ) ) ) ).
% enumerate_Suc_eq
thf(fact_6093_map__eq__imp__length__eq,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,Xs2: list @ B,G3: C > A,Ys: list @ C] :
( ( ( map @ B @ A @ F3 @ Xs2 )
= ( map @ C @ A @ G3 @ Ys ) )
=> ( ( size_size @ ( list @ B ) @ Xs2 )
= ( size_size @ ( list @ C ) @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_6094_list_Omap__cong,axiom,
! [B: $tType,A: $tType,X3: list @ A,Ya: list @ A,F3: A > B,G3: A > B] :
( ( X3 = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ Ya ) )
=> ( ( F3 @ Z3 )
= ( G3 @ Z3 ) ) )
=> ( ( map @ A @ B @ F3 @ X3 )
= ( map @ A @ B @ G3 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_6095_list_Omap__cong0,axiom,
! [B: $tType,A: $tType,X3: list @ A,F3: A > B,G3: A > B] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ X3 ) )
=> ( ( F3 @ Z3 )
= ( G3 @ Z3 ) ) )
=> ( ( map @ A @ B @ F3 @ X3 )
= ( map @ A @ B @ G3 @ X3 ) ) ) ).
% list.map_cong0
thf(fact_6096_list_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X3: list @ A,Xa2: list @ A,F3: A > B,Fa: A > B] :
( ! [Z3: A,Za2: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ X3 ) )
=> ( ( member @ A @ Za2 @ ( set2 @ A @ Xa2 ) )
=> ( ( ( F3 @ Z3 )
= ( Fa @ Za2 ) )
=> ( Z3 = Za2 ) ) ) )
=> ( ( ( map @ A @ B @ F3 @ X3 )
= ( map @ A @ B @ Fa @ Xa2 ) )
=> ( X3 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_6097_map__ext,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,F3: A > B,G3: A > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ( F3 @ X5 )
= ( G3 @ X5 ) ) )
=> ( ( map @ A @ B @ F3 @ Xs2 )
= ( map @ A @ B @ G3 @ Xs2 ) ) ) ).
% map_ext
thf(fact_6098_map__idI,axiom,
! [A: $tType,Xs2: list @ A,F3: A > A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ( F3 @ X5 )
= X5 ) )
=> ( ( map @ A @ A @ F3 @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_6099_map__cong,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ A,F3: A > B,G3: A > B] :
( ( Xs2 = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
=> ( ( F3 @ X5 )
= ( G3 @ X5 ) ) )
=> ( ( map @ A @ B @ F3 @ Xs2 )
= ( map @ A @ B @ G3 @ Ys ) ) ) ) ).
% map_cong
thf(fact_6100_ex__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ B,F3: A > B] :
( ( ? [Xs: list @ A] :
( Ys
= ( map @ A @ B @ F3 @ Xs ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Ys ) )
=> ? [Y3: A] :
( X4
= ( F3 @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_6101_image__set,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs2: list @ B] :
( ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs2 ) )
= ( set2 @ A @ ( map @ B @ A @ F3 @ Xs2 ) ) ) ).
% image_set
thf(fact_6102_map__replicate__const,axiom,
! [B: $tType,A: $tType,K2: A,Lst: list @ B] :
( ( map @ B @ A
@ ^ [X4: B] : K2
@ Lst )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K2 ) ) ).
% map_replicate_const
thf(fact_6103_inj__on__map__eq__map,axiom,
! [B: $tType,A: $tType,F3: A > B,Xs2: list @ A,Ys: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ Ys ) ) )
=> ( ( ( map @ A @ B @ F3 @ Xs2 )
= ( map @ A @ B @ F3 @ Ys ) )
= ( Xs2 = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_6104_map__inj__on,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs2: list @ B,Ys: list @ B] :
( ( ( map @ B @ A @ F3 @ Xs2 )
= ( map @ B @ A @ F3 @ Ys ) )
=> ( ( inj_on @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs2 ) @ ( set2 @ B @ Ys ) ) )
=> ( Xs2 = Ys ) ) ) ).
% map_inj_on
thf(fact_6105_distinct__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs2: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ Xs2 ) )
= ( ( distinct @ B @ Xs2 )
& ( inj_on @ B @ A @ F3 @ ( set2 @ B @ Xs2 ) ) ) ) ).
% distinct_map
thf(fact_6106_map__of__eqI,axiom,
! [B: $tType,A: $tType,Xs2: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs2 ) )
= ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs2 ) ) )
=> ( ( map_of @ A @ B @ Xs2 @ X5 )
= ( map_of @ A @ B @ Ys @ X5 ) ) )
=> ( ( map_of @ A @ B @ Xs2 )
= ( map_of @ A @ B @ Ys ) ) ) ) ).
% map_of_eqI
thf(fact_6107_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs2: list @ A,Ys: list @ B,Zs2: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs2 @ ( zip @ B @ C @ Ys @ Zs2 ) )
= ( map @ ( product_prod @ ( product_prod @ A @ B ) @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ ( product_prod @ B @ C ) ) )
@ ^ [X4: A,Y3: B,Z4: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X4 @ ( product_Pair @ B @ C @ Y3 @ Z4 ) ) ) )
@ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs2 @ Ys ) @ Zs2 ) ) ) ).
% zip_assoc
thf(fact_6108_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs2: list @ A,Ys: list @ B,Zs2: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs2 @ ( zip @ B @ C @ Ys @ Zs2 ) )
= ( map @ ( product_prod @ B @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ B @ ( product_prod @ A @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [Y3: B] :
( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [X4: A,Z4: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X4 @ ( product_Pair @ B @ C @ Y3 @ Z4 ) ) ) )
@ ( zip @ B @ ( product_prod @ A @ C ) @ Ys @ ( zip @ A @ C @ Xs2 @ Zs2 ) ) ) ) ).
% zip_left_commute
thf(fact_6109_zip__commute,axiom,
! [B: $tType,A: $tType] :
( ( zip @ A @ B )
= ( ^ [Xs: list @ A,Ys3: list @ B] :
( map @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X4: B,Y3: A] : ( product_Pair @ A @ B @ Y3 @ X4 ) )
@ ( zip @ B @ A @ Ys3 @ Xs ) ) ) ) ).
% zip_commute
thf(fact_6110_zip__eq__conv,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,Zs2: list @ ( product_prod @ A @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( zip @ A @ B @ Xs2 @ Ys )
= Zs2 )
= ( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs2 )
= Xs2 )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs2 )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_6111_distinct__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Xs2: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs2 ) ) )
= ( ~ ( member @ A @ ( F3 @ X3 ) @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs2 ) ) )
& ( distinct @ A @ ( map @ B @ A @ F3 @ Xs2 ) ) ) ) ) ).
% distinct_insort_key
thf(fact_6112_map__removeAll__inj__on,axiom,
! [B: $tType,A: $tType,F3: A > B,X3: A,Xs2: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( insert2 @ A @ X3 @ ( set2 @ A @ Xs2 ) ) )
=> ( ( map @ A @ B @ F3 @ ( removeAll @ A @ X3 @ Xs2 ) )
= ( removeAll @ B @ ( F3 @ X3 ) @ ( map @ A @ B @ F3 @ Xs2 ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_6113_eq__key__imp__eq__value,axiom,
! [A: $tType,B: $tType,Xs2: list @ ( product_prod @ A @ B ),K2: A,V1: B,V22: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs2 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V1 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs2 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V22 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs2 ) )
=> ( V1 = V22 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_6114_inj__on__mapI,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ ( list @ A )] :
( ( inj_on @ A @ B @ F3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A6 ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F3 ) @ A6 ) ) ).
% inj_on_mapI
thf(fact_6115_map__of__zip__map,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,F3: A > B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ ( map @ A @ B @ F3 @ Xs2 ) ) )
= ( ^ [X4: A] : ( if @ ( option @ B ) @ ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) ) @ ( some @ B @ ( F3 @ X4 ) ) @ ( none @ B ) ) ) ) ).
% map_of_zip_map
thf(fact_6116_map__fst__zip__take,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) )
= ( take @ A @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) ) @ Xs2 ) ) ).
% map_fst_zip_take
thf(fact_6117_map__snd__zip__take,axiom,
! [B: $tType,A: $tType,Xs2: list @ B,Ys: list @ A] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( zip @ B @ A @ Xs2 @ Ys ) )
= ( take @ A @ ( ord_min @ nat @ ( size_size @ ( list @ B ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ Ys ) ) ).
% map_snd_zip_take
thf(fact_6118_map__of__map,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,Xs2: list @ ( product_prod @ A @ C )] :
( ( map_of @ A @ B
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [K3: A,V5: C] : ( product_Pair @ A @ B @ K3 @ ( F3 @ V5 ) ) )
@ Xs2 ) )
= ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F3 ) @ ( map_of @ A @ C @ Xs2 ) ) ) ).
% map_of_map
thf(fact_6119_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,T2: list @ ( product_prod @ A @ C ),K2: A,X3: C] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map_of @ A @ C @ T2 @ K2 )
= ( some @ C @ X3 ) )
=> ( ( map_of @ B @ C
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C )
@ ( product_case_prod @ A @ C @ ( product_prod @ B @ C )
@ ^ [K3: A] : ( product_Pair @ B @ C @ ( F3 @ K3 ) ) )
@ T2 )
@ ( F3 @ K2 ) )
= ( some @ C @ X3 ) ) ) ) ).
% map_of_mapk_SomeI
thf(fact_6120_set__map__of__compr,axiom,
! [B: $tType,A: $tType,Xs2: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs2 ) )
=> ( ( set2 @ ( product_prod @ A @ B ) @ Xs2 )
= ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K3: A,V5: B] :
( ( map_of @ A @ B @ Xs2 @ K3 )
= ( some @ B @ V5 ) ) ) ) ) ) ).
% set_map_of_compr
thf(fact_6121_set__relcomp,axiom,
! [B: $tType,C: $tType,A: $tType,Xys2: list @ ( product_prod @ A @ C ),Yzs: list @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( set2 @ ( product_prod @ A @ C ) @ Xys2 ) @ ( set2 @ ( product_prod @ C @ B ) @ Yzs ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ A @ C ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Xy: product_prod @ A @ C] :
( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ C @ B ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Yz: product_prod @ C @ B] :
( if @ ( list @ ( product_prod @ A @ B ) )
@ ( ( product_snd @ A @ C @ Xy )
= ( product_fst @ C @ B @ Yz ) )
@ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Xy ) @ ( product_snd @ C @ B @ Yz ) ) @ ( nil @ ( product_prod @ A @ B ) ) )
@ ( nil @ ( product_prod @ A @ B ) ) )
@ Yzs ) )
@ Xys2 ) ) ) ) ).
% set_relcomp
thf(fact_6122_zip__Cons1,axiom,
! [A: $tType,B: $tType,X3: A,Xs2: list @ A,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X3 @ Xs2 ) @ Ys )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Y3: B,Ys3: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( zip @ A @ B @ Xs2 @ Ys3 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_6123_zip__Cons,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Y: B,Ys: list @ B] :
( ( zip @ A @ B @ Xs2 @ ( cons @ B @ Y @ Ys ) )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ A @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Z4: A,Zs3: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z4 @ Y ) @ ( zip @ A @ B @ Zs3 @ Ys ) )
@ Xs2 ) ) ).
% zip_Cons
thf(fact_6124_zip__same__conv__map,axiom,
! [A: $tType,Xs2: list @ A] :
( ( zip @ A @ A @ Xs2 @ Xs2 )
= ( map @ A @ ( product_prod @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Xs2 ) ) ).
% zip_same_conv_map
thf(fact_6125_product__concat__map,axiom,
! [B: $tType,A: $tType] :
( ( product @ A @ B )
= ( ^ [Xs: list @ A,Ys3: list @ B] :
( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 ) @ Ys3 )
@ Xs ) ) ) ) ).
% product_concat_map
thf(fact_6126_n__lists_Osimps_I2_J,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( n_lists @ A @ ( suc @ N ) @ Xs2 )
= ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ^ [Ys3: list @ A] :
( map @ A @ ( list @ A )
@ ^ [Y3: A] : ( cons @ A @ Y3 @ Ys3 )
@ Xs2 )
@ ( n_lists @ A @ N @ Xs2 ) ) ) ) ).
% n_lists.simps(2)
thf(fact_6127_distinct__set__subseqs,axiom,
! [A: $tType,Xs2: list @ A] :
( ( distinct @ A @ Xs2 )
=> ( distinct @ ( set @ A ) @ ( map @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( subseqs @ A @ Xs2 ) ) ) ) ).
% distinct_set_subseqs
thf(fact_6128_zip__replicate1,axiom,
! [A: $tType,B: $tType,N: nat,X3: A,Ys: list @ B] :
( ( zip @ A @ B @ ( replicate @ A @ N @ X3 ) @ Ys )
= ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ ( take @ B @ N @ Ys ) ) ) ).
% zip_replicate1
thf(fact_6129_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,N: nat,Y: B] :
( ( zip @ A @ B @ Xs2 @ ( replicate @ B @ N @ Y ) )
= ( map @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ Y )
@ ( take @ A @ N @ Xs2 ) ) ) ).
% zip_replicate2
thf(fact_6130_Id__on__set,axiom,
! [A: $tType,Xs2: list @ A] :
( ( id_on @ A @ ( set2 @ A @ Xs2 ) )
= ( set2 @ ( product_prod @ A @ A )
@ ( map @ A @ ( product_prod @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Xs2 ) ) ) ).
% Id_on_set
thf(fact_6131_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X3: A,Xs2: list @ A,Ys: list @ B] :
( ( product @ A @ B @ ( cons @ A @ X3 @ Xs2 ) @ Ys )
= ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ Ys ) @ ( product @ A @ B @ Xs2 @ Ys ) ) ) ).
% product.simps(2)
thf(fact_6132_map__of__map__keys,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,M2: A > ( option @ B )] :
( ( ( set2 @ A @ Xs2 )
= ( dom @ A @ B @ M2 ) )
=> ( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( the2 @ B @ ( M2 @ K3 ) ) )
@ Xs2 ) )
= M2 ) ) ).
% map_of_map_keys
thf(fact_6133_map__of__map__restrict,axiom,
! [B: $tType,A: $tType,F3: A > B,Ks2: list @ A] :
( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( F3 @ K3 ) )
@ Ks2 ) )
= ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F3 ) @ ( set2 @ A @ Ks2 ) ) ) ).
% map_of_map_restrict
thf(fact_6134_product__code,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B] :
( ( product_product @ A @ B @ ( set2 @ A @ Xs2 ) @ ( set2 @ B @ Ys ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 ) @ Ys )
@ Xs2 ) ) ) ) ).
% product_code
thf(fact_6135_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_6136_drop__upt,axiom,
! [M2: nat,I: nat,J: nat] :
( ( drop @ nat @ M2 @ ( upt @ I @ J ) )
= ( upt @ ( plus_plus @ nat @ I @ M2 ) @ J ) ) ).
% drop_upt
thf(fact_6137_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size @ ( list @ nat ) @ ( upt @ I @ J ) )
= ( minus_minus @ nat @ J @ I ) ) ).
% length_upt
thf(fact_6138_take__upt,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ M2 ) @ N )
=> ( ( take @ nat @ M2 @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus @ nat @ I @ M2 ) ) ) ) ).
% take_upt
thf(fact_6139_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( upt @ I @ J )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_6140_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_6141_nth__upt,axiom,
! [I: nat,K2: nat,J: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ K2 ) @ J )
=> ( ( nth @ nat @ ( upt @ I @ J ) @ K2 )
= ( plus_plus @ nat @ I @ K2 ) ) ) ).
% nth_upt
thf(fact_6142_map__fst__enumerate,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs2 ) )
= ( upt @ N @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% map_fst_enumerate
thf(fact_6143_upt__rec__numeral,axiom,
! [M2: num,N: num] :
( ( ( ord_less @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) )
= ( cons @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral @ nat @ M2 ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) )
& ( ~ ( ord_less @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) )
= ( nil @ nat ) ) ) ) ).
% upt_rec_numeral
thf(fact_6144_map__Suc__upt,axiom,
! [M2: nat,N: nat] :
( ( map @ nat @ nat @ suc @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_6145_upt__conv__Cons__Cons,axiom,
! [M2: nat,N: nat,Ns: list @ nat,Q3: nat] :
( ( ( cons @ nat @ M2 @ ( cons @ nat @ N @ Ns ) )
= ( upt @ M2 @ Q3 ) )
= ( ( cons @ nat @ N @ Ns )
= ( upt @ ( suc @ M2 ) @ Q3 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_6146_atLeastAtMost__upt,axiom,
( ( set_or1337092689740270186AtMost @ nat )
= ( ^ [N3: nat,M5: nat] : ( set2 @ nat @ ( upt @ N3 @ ( suc @ M5 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_6147_greaterThanAtMost__upt,axiom,
( ( set_or3652927894154168847AtMost @ nat )
= ( ^ [N3: nat,M5: nat] : ( set2 @ nat @ ( upt @ ( suc @ N3 ) @ ( suc @ M5 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_6148_map__add__upt,axiom,
! [N: nat,M2: nat] :
( ( map @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ I4 @ N )
@ ( upt @ ( zero_zero @ nat ) @ M2 ) )
= ( upt @ N @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% map_add_upt
thf(fact_6149_greaterThanLessThan__upt,axiom,
( ( set_or5935395276787703475ssThan @ nat )
= ( ^ [N3: nat,M5: nat] : ( set2 @ nat @ ( upt @ ( suc @ N3 ) @ M5 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_6150_enumerate__map__upt,axiom,
! [A: $tType,N: nat,F3: nat > A,M2: nat] :
( ( enumerate @ A @ N @ ( map @ nat @ A @ F3 @ ( upt @ N @ M2 ) ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [K3: nat] : ( product_Pair @ nat @ A @ K3 @ ( F3 @ K3 ) )
@ ( upt @ N @ M2 ) ) ) ).
% enumerate_map_upt
thf(fact_6151_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) ) ) ) ).
% atMost_upto
thf(fact_6152_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_6153_enumerate__eq__zip,axiom,
! [A: $tType] :
( ( enumerate @ A )
= ( ^ [N3: nat,Xs: list @ A] : ( zip @ nat @ A @ ( upt @ N3 @ ( plus_plus @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) ) ) @ Xs ) ) ) ).
% enumerate_eq_zip
thf(fact_6154_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_6155_map__decr__upt,axiom,
! [M2: nat,N: nat] :
( ( map @ nat @ nat
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( upt @ M2 @ N ) ) ).
% map_decr_upt
thf(fact_6156_map__nth,axiom,
! [A: $tType,Xs2: list @ A] :
( ( map @ nat @ A @ ( nth @ A @ Xs2 ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
= Xs2 ) ).
% map_nth
thf(fact_6157_upt__add__eq__append,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus @ nat @ J @ K2 ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K2 ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_6158_nth__map__upt,axiom,
! [A: $tType,I: nat,N: nat,M2: nat,F3: nat > A] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N @ M2 ) )
=> ( ( nth @ A @ ( map @ nat @ A @ F3 @ ( upt @ M2 @ N ) ) @ I )
= ( F3 @ ( plus_plus @ nat @ M2 @ I ) ) ) ) ).
% nth_map_upt
thf(fact_6159_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X3: nat,Xs2: list @ nat] :
( ( ( upt @ I @ J )
= ( cons @ nat @ X3 @ Xs2 ) )
= ( ( ord_less @ nat @ I @ J )
& ( I = X3 )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
= Xs2 ) ) ) ).
% upt_eq_Cons_conv
thf(fact_6160_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_6161_enumerate__replicate__eq,axiom,
! [A: $tType,N: nat,M2: nat,A3: A] :
( ( enumerate @ A @ N @ ( replicate @ A @ M2 @ A3 ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [Q4: nat] : ( product_Pair @ nat @ A @ Q4 @ A3 )
@ ( upt @ N @ ( plus_plus @ nat @ N @ M2 ) ) ) ) ).
% enumerate_replicate_eq
thf(fact_6162_map__upt__eqI,axiom,
! [A: $tType,Xs2: list @ A,N: nat,M2: nat,F3: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( minus_minus @ nat @ N @ M2 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I3 )
= ( F3 @ ( plus_plus @ nat @ M2 @ I3 ) ) ) )
=> ( ( map @ nat @ A @ F3 @ ( upt @ M2 @ N ) )
= Xs2 ) ) ) ).
% map_upt_eqI
thf(fact_6163_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_6164_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_6165_transpose__rectangle,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),N: nat] :
( ( ( Xs2
= ( nil @ ( list @ A ) ) )
=> ( N
= ( zero_zero @ nat ) ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs2 @ I3 ) )
= N ) )
=> ( ( transpose @ A @ Xs2 )
= ( map @ nat @ ( list @ A )
@ ^ [I4: nat] :
( map @ nat @ A
@ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs2 @ J3 ) @ I4 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_6166_length__product__lists,axiom,
! [B: $tType,Xss: list @ ( list @ B )] :
( ( size_size @ ( list @ ( list @ B ) ) @ ( product_lists @ B @ Xss ) )
= ( foldr @ nat @ nat @ ( times_times @ nat ) @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) @ ( one_one @ nat ) ) ) ).
% length_product_lists
thf(fact_6167_foldr__cong,axiom,
! [B: $tType,A: $tType,A3: A,B2: A,L: list @ B,K2: list @ B,F3: B > A > A,G3: B > A > A] :
( ( A3 = B2 )
=> ( ( L = K2 )
=> ( ! [A5: A,X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ L ) )
=> ( ( F3 @ X5 @ A5 )
= ( G3 @ X5 @ A5 ) ) )
=> ( ( foldr @ B @ A @ F3 @ L @ A3 )
= ( foldr @ B @ A @ G3 @ K2 @ B2 ) ) ) ) ) ).
% foldr_cong
thf(fact_6168_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A8: A,Xs: list @ B] :
( foldr @ B @ A
@ ^ [X4: B,B8: A] : ( plus_plus @ A @ ( F4 @ X4 ) @ ( times_times @ A @ A8 @ B8 ) )
@ Xs
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_6169_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs2: list @ A,X6: set @ A,F3: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ X6 )
=> ( ( finite_finite2 @ A @ X6 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs2 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( times_times @ nat @ ( count_list @ A @ Xs2 @ X4 ) @ ( F3 @ X4 ) )
@ X6 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_6170_nth__transpose,axiom,
! [A: $tType,I: nat,Xs2: list @ ( list @ A )] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs2 ) @ I )
= ( map @ ( list @ A ) @ A
@ ^ [Xs: list @ A] : ( nth @ A @ Xs @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs2 ) ) ) ) ).
% nth_transpose
thf(fact_6171_filter__True,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X5 ) )
=> ( ( filter2 @ A @ P @ Xs2 )
= Xs2 ) ) ).
% filter_True
thf(fact_6172_set__filter,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( set2 @ A @ ( filter2 @ A @ P @ Xs2 ) )
= ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
& ( P @ X4 ) ) ) ) ).
% set_filter
thf(fact_6173_sum__list__eq__0__iff,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [Ns: list @ A] :
( ( ( groups8242544230860333062m_list @ A @ Ns )
= ( zero_zero @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ns ) )
=> ( X4
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_6174_sum__list_OCons,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [X3: A,Xs2: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( cons @ A @ X3 @ Xs2 ) )
= ( plus_plus @ A @ X3 @ ( groups8242544230860333062m_list @ A @ Xs2 ) ) ) ) ).
% sum_list.Cons
thf(fact_6175_filter__False,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ~ ( P @ X5 ) )
=> ( ( filter2 @ A @ P @ Xs2 )
= ( nil @ A ) ) ) ).
% filter_False
thf(fact_6176_sum__list__append,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs2: list @ A,Ys: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( append @ A @ Xs2 @ Ys ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs2 ) @ ( groups8242544230860333062m_list @ A @ Ys ) ) ) ) ).
% sum_list_append
thf(fact_6177_length__filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: B > A,Xs2: list @ B] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ ( map @ B @ A @ F3 @ Xs2 ) ) )
= ( size_size @ ( list @ B ) @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F3 ) @ Xs2 ) ) ) ).
% length_filter_map
thf(fact_6178_sum__list__upt,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups8242544230860333062m_list @ nat @ ( upt @ M2 @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum_list_upt
thf(fact_6179_sum__list__filter__le__nat,axiom,
! [A: $tType,F3: A > nat,P: A > $o,Xs2: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ ( filter2 @ A @ P @ Xs2 ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs2 ) ) ) ).
% sum_list_filter_le_nat
thf(fact_6180_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs2: list @ B,P: B > $o,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ Xs2 ) )
=> ( ~ ( P @ X5 )
=> ( ( F3 @ X5 )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P @ Xs2 ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs2 ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_6181_filter__empty__conv,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( ( filter2 @ A @ P @ Xs2 )
= ( nil @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ~ ( P @ X4 ) ) ) ) ).
% filter_empty_conv
thf(fact_6182_empty__filter__conv,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( ( nil @ A )
= ( filter2 @ A @ P @ Xs2 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ~ ( P @ X4 ) ) ) ) ).
% empty_filter_conv
thf(fact_6183_inter__set__filter,axiom,
! [A: $tType,A6: set @ A,Xs2: list @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ ( set2 @ A @ Xs2 ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A6 )
@ Xs2 ) ) ) ).
% inter_set_filter
thf(fact_6184_filter__id__conv,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( ( filter2 @ A @ P @ Xs2 )
= Xs2 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X4 ) ) ) ) ).
% filter_id_conv
thf(fact_6185_filter__cong,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] :
( ( Xs2 = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( filter2 @ A @ P @ Xs2 )
= ( filter2 @ A @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_6186_member__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ X3 @ ( groups8242544230860333062m_list @ A @ Xs2 ) ) ) ) ).
% member_le_sum_list
thf(fact_6187_filter__is__subset,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( filter2 @ A @ P @ Xs2 ) ) @ ( set2 @ A @ Xs2 ) ) ).
% filter_is_subset
thf(fact_6188_filter__set,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( filter3 @ A @ P @ ( set2 @ A @ Xs2 ) )
= ( set2 @ A @ ( filter2 @ A @ P @ Xs2 ) ) ) ).
% filter_set
thf(fact_6189_replicate__length__filter,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( replicate @ A
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y5: A,Z: A] : Y5 = Z
@ X3 )
@ Xs2 ) )
@ X3 )
= ( filter2 @ A
@ ( ^ [Y5: A,Z: A] : Y5 = Z
@ X3 )
@ Xs2 ) ) ).
% replicate_length_filter
thf(fact_6190_sum__length__filter__compl,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs2 ) )
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ^ [X4: A] :
~ ( P @ X4 )
@ Xs2 ) ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% sum_length_filter_compl
thf(fact_6191_length__filter__le,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_filter_le
thf(fact_6192_sum__list__addf,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: B > A,G3: B > A,Xs2: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ Xs2 ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs2 ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G3 @ Xs2 ) ) ) ) ) ).
% sum_list_addf
thf(fact_6193_length__filter__less,axiom,
! [A: $tType,X3: A,Xs2: list @ A,P: A > $o] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ~ ( P @ X3 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% length_filter_less
thf(fact_6194_Cons__eq__filterD,axiom,
! [A: $tType,X3: A,Xs2: list @ A,P: A > $o,Ys: list @ A] :
( ( ( cons @ A @ X3 @ Xs2 )
= ( filter2 @ A @ P @ Ys ) )
=> ? [Us3: list @ A,Vs3: list @ A] :
( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ X3 @ Vs3 ) ) )
& ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Us3 ) )
=> ~ ( P @ X ) )
& ( P @ X3 )
& ( Xs2
= ( filter2 @ A @ P @ Vs3 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_6195_filter__eq__ConsD,axiom,
! [A: $tType,P: A > $o,Ys: list @ A,X3: A,Xs2: list @ A] :
( ( ( filter2 @ A @ P @ Ys )
= ( cons @ A @ X3 @ Xs2 ) )
=> ? [Us3: list @ A,Vs3: list @ A] :
( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ X3 @ Vs3 ) ) )
& ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Us3 ) )
=> ~ ( P @ X ) )
& ( P @ X3 )
& ( Xs2
= ( filter2 @ A @ P @ Vs3 ) ) ) ) ).
% filter_eq_ConsD
thf(fact_6196_Cons__eq__filter__iff,axiom,
! [A: $tType,X3: A,Xs2: list @ A,P: A > $o,Ys: list @ A] :
( ( ( cons @ A @ X3 @ Xs2 )
= ( filter2 @ A @ P @ Ys ) )
= ( ? [Us2: list @ A,Vs2: list @ A] :
( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ X3 @ Vs2 ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Us2 ) )
=> ~ ( P @ X4 ) )
& ( P @ X3 )
& ( Xs2
= ( filter2 @ A @ P @ Vs2 ) ) ) ) ) ).
% Cons_eq_filter_iff
thf(fact_6197_filter__eq__Cons__iff,axiom,
! [A: $tType,P: A > $o,Ys: list @ A,X3: A,Xs2: list @ A] :
( ( ( filter2 @ A @ P @ Ys )
= ( cons @ A @ X3 @ Xs2 ) )
= ( ? [Us2: list @ A,Vs2: list @ A] :
( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ X3 @ Vs2 ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Us2 ) )
=> ~ ( P @ X4 ) )
& ( P @ X3 )
& ( Xs2
= ( filter2 @ A @ P @ Vs2 ) ) ) ) ) ).
% filter_eq_Cons_iff
thf(fact_6198_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,Y: A,Xs2: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( insert2 @ A @ Y @ ( set2 @ A @ Xs2 ) ) )
=> ( ( filter2 @ A
@ ^ [X4: A] :
( ( F3 @ Y )
= ( F3 @ X4 ) )
@ Xs2 )
= ( filter2 @ A
@ ( ^ [Y5: A,Z: A] : Y5 = Z
@ Y )
@ Xs2 ) ) ) ).
% inj_on_filter_key_eq
thf(fact_6199_sum__list__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs2: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ X5 @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs2 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_nonpos
thf(fact_6200_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs2: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 ) )
=> ( ( ( groups8242544230860333062m_list @ A @ Xs2 )
= ( zero_zero @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( X4
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_6201_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs2: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs2 ) ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_6202_sum__list__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [Xs2: list @ A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( groups8242544230860333062m_list @ A @ Xs2 ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ A @ A @ ( abs_abs @ A ) @ Xs2 ) ) ) ) ).
% sum_list_abs
thf(fact_6203_transpose__aux__max,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Xss: list @ ( list @ B )] :
( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [Xs: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs ) )
@ Xss
@ ( zero_zero @ nat ) ) )
= ( suc
@ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [X4: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X4 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter2 @ ( list @ B )
@ ^ [Ys3: list @ B] :
( Ys3
!= ( nil @ B ) )
@ Xss )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_6204_filter__in__nths,axiom,
! [A: $tType,Xs2: list @ A,S: set @ nat] :
( ( distinct @ A @ Xs2 )
=> ( ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( set2 @ A @ ( nths @ A @ Xs2 @ S ) ) )
@ Xs2 )
= ( nths @ A @ Xs2 @ S ) ) ) ).
% filter_in_nths
thf(fact_6205_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( ^ [Xs: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ) ).
% sum_list.eq_foldr
thf(fact_6206_transpose__max__length,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( foldr @ ( list @ A ) @ nat
@ ^ [Xs: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs ) )
@ ( transpose @ A @ Xs2 )
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [X4: list @ A] :
( X4
!= ( nil @ A ) )
@ Xs2 ) ) ) ).
% transpose_max_length
thf(fact_6207_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs2: list @ A,F3: A > B,G3: A > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) )
=> ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F3 @ Xs2 ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G3 @ Xs2 ) ) ) ) ) ).
% sum_list_mono
thf(fact_6208_length__concat,axiom,
! [B: $tType,Xss: list @ ( list @ B )] :
( ( size_size @ ( list @ B ) @ ( concat @ B @ Xss ) )
= ( groups8242544230860333062m_list @ nat @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) ) ) ).
% length_concat
thf(fact_6209_distinct__sum__list__conv__Sum,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs2: list @ A] :
( ( distinct @ A @ Xs2 )
=> ( ( groups8242544230860333062m_list @ A @ Xs2 )
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : X4
@ ( set2 @ A @ Xs2 ) ) ) ) ) ).
% distinct_sum_list_conv_Sum
thf(fact_6210_set__minus__filter__out,axiom,
! [A: $tType,Xs2: list @ A,Y: A] :
( ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X4: A] : X4 != Y
@ Xs2 ) ) ) ).
% set_minus_filter_out
thf(fact_6211_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs2 @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
@ Zs2 )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_6212_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs2 @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
@ Zs2 )
= Xs2 ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_6213_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs2 @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
@ Zs2 )
= Xs2 ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_6214_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs2 @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
@ Zs2 )
= Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_6215_filter__eq__nths,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P4: A > $o,Xs: list @ A] :
( nths @ A @ Xs
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P4 @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_6216_length__filter__conv__card,axiom,
! [A: $tType,P2: A > $o,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P2 @ Xs2 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P2 @ ( nth @ A @ Xs2 @ I4 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_6217_elem__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [K2: nat,Ns: list @ A] :
( ( ord_less @ nat @ K2 @ ( size_size @ ( list @ A ) @ Ns ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Ns @ K2 ) @ ( groups8242544230860333062m_list @ A @ Ns ) ) ) ) ).
% elem_le_sum_list
thf(fact_6218_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( strict9044650504122735259up_add @ B ) )
=> ! [Xs2: list @ A,F3: A > B,G3: A > B] :
( ( Xs2
!= ( nil @ A ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) )
=> ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F3 @ Xs2 ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G3 @ Xs2 ) ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_6219_sum_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs2: list @ B,G3: B > A] :
( ( distinct @ B @ Xs2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( set2 @ B @ Xs2 ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G3 @ Xs2 ) ) ) ) ) ).
% sum.distinct_set_conv_list
thf(fact_6220_sum__list__distinct__conv__sum__set,axiom,
! [C: $tType,B: $tType] :
( ( comm_monoid_add @ C )
=> ! [Xs2: list @ B,F3: B > C] :
( ( distinct @ B @ Xs2 )
=> ( ( groups8242544230860333062m_list @ C @ ( map @ B @ C @ F3 @ Xs2 ) )
= ( groups7311177749621191930dd_sum @ B @ C @ F3 @ ( set2 @ B @ Xs2 ) ) ) ) ) ).
% sum_list_distinct_conv_sum_set
thf(fact_6221_sum__list__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [X3: B,Xs2: list @ B,F3: B > A] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs2 ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs2 ) )
= ( plus_plus @ A @ ( F3 @ X3 ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( remove1 @ B @ X3 @ Xs2 ) ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_6222_sum__code,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A,Xs2: list @ B] :
( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( set2 @ B @ Xs2 ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G3 @ ( remdups @ B @ Xs2 ) ) ) ) ) ).
% sum_code
thf(fact_6223_size__list__conv__sum__list,axiom,
! [B: $tType] :
( ( size_list @ B )
= ( ^ [F4: B > nat,Xs: list @ B] : ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ B @ nat @ F4 @ Xs ) ) @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ).
% size_list_conv_sum_list
thf(fact_6224_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R2: B,Xs2: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X4: C] : R2
@ Xs2 ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs2 ) ) @ R2 ) ) ) ).
% sum_list_triv
thf(fact_6225_sum__list__Suc,axiom,
! [A: $tType,F3: A > nat,Xs2: list @ A] :
( ( groups8242544230860333062m_list @ nat
@ ( map @ A @ nat
@ ^ [X4: A] : ( suc @ ( F3 @ X4 ) )
@ Xs2 ) )
= ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% sum_list_Suc
thf(fact_6226_distinct__length__filter,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ( distinct @ A @ Xs2 )
=> ( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs2 ) )
= ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( set2 @ A @ Xs2 ) ) ) ) ) ).
% distinct_length_filter
thf(fact_6227_sum__list__sum__nth,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ( ( groups8242544230860333062m_list @ B )
= ( ^ [Xs: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ) ) ).
% sum_list_sum_nth
thf(fact_6228_card__length__sum__list__rec,axiom,
! [M2: nat,N5: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M2 )
=> ( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M2 )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N5 ) ) ) )
= ( plus_plus @ nat
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N5 ) ) ) )
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M2 )
& ( ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ L2 ) @ ( one_one @ nat ) )
= N5 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_6229_card__length__sum__list,axiom,
! [M2: nat,N5: nat] :
( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M2 )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N5 ) ) ) )
= ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N5 @ M2 ) @ ( one_one @ nat ) ) @ N5 ) ) ).
% card_length_sum_list
thf(fact_6230_sum__list__map__eq__sum__count,axiom,
! [A: $tType,F3: A > nat,Xs2: list @ A] :
( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs2 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( times_times @ nat @ ( count_list @ A @ Xs2 @ X4 ) @ ( F3 @ X4 ) )
@ ( set2 @ A @ Xs2 ) ) ) ).
% sum_list_map_eq_sum_count
thf(fact_6231_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K2: nat,Xs2: list @ A,X3: A] :
( ( ord_less @ nat @ K2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs2 @ K2 @ X3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs2 ) @ X3 ) @ ( nth @ A @ Xs2 @ K2 ) ) ) ) ) ).
% sum_list_update
thf(fact_6232_length__transpose,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) )
= ( foldr @ ( list @ A ) @ nat
@ ^ [Xs: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs ) )
@ Xs2
@ ( zero_zero @ nat ) ) ) ).
% length_transpose
thf(fact_6233_map__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter @ A @ B )
= ( ^ [F4: A > ( option @ B ),Xs: list @ A] :
( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F4 )
@ ( filter2 @ A
@ ^ [X4: A] :
( ( F4 @ X4 )
!= ( none @ B ) )
@ Xs ) ) ) ) ).
% map_filter_def
thf(fact_6234_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F3: B > A,P: B > $o,Xs2: list @ B] :
( ( map @ B @ A @ F3 @ ( filter2 @ B @ P @ Xs2 ) )
= ( map_filter @ B @ A
@ ^ [X4: B] : ( if @ ( option @ A ) @ ( P @ X4 ) @ ( some @ A @ ( F3 @ X4 ) ) @ ( none @ A ) )
@ Xs2 ) ) ).
% map_filter_map_filter
thf(fact_6235_map__of__filter__in,axiom,
! [B: $tType,A: $tType,Xs2: list @ ( product_prod @ B @ A ),K2: B,Z2: A,P: B > A > $o] :
( ( ( map_of @ B @ A @ Xs2 @ K2 )
= ( some @ A @ Z2 ) )
=> ( ( P @ K2 @ Z2 )
=> ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) @ Xs2 ) @ K2 )
= ( some @ A @ Z2 ) ) ) ) ).
% map_of_filter_in
thf(fact_6236_nths__shift__lemma__Suc,axiom,
! [A: $tType,P: nat > $o,Xs2: list @ A,Is: list @ nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P5: product_prod @ A @ nat] : ( P @ ( suc @ ( product_snd @ A @ nat @ P5 ) ) )
@ ( zip @ A @ nat @ Xs2 @ Is ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P5: product_prod @ A @ nat] : ( P @ ( product_snd @ A @ nat @ P5 ) )
@ ( zip @ A @ nat @ Xs2 @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ).
% nths_shift_lemma_Suc
thf(fact_6237_nths__shift__lemma,axiom,
! [A: $tType,A6: set @ nat,Xs2: 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 ) @ A6 )
@ ( zip @ A @ nat @ Xs2 @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) )
= ( 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 ) @ A6 )
@ ( zip @ A @ nat @ Xs2 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_6238_nths__def,axiom,
! [A: $tType] :
( ( nths @ A )
= ( ^ [Xs: list @ A,A7: 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 ) @ A7 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ) ).
% nths_def
thf(fact_6239_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F3: nat > B,Ns: list @ nat] :
( ! [X5: nat,Y4: nat] :
( ( ord_less_eq @ nat @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) )
=> ( ( 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_6240_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [A3: B,Xs2: list @ B,F3: B > A] :
( ( member @ B @ A3 @ ( set2 @ B @ Xs2 ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs2 ) )
=> ( ( ( hd @ B
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F3 @ A3 )
= ( F3 @ X4 ) )
@ Xs2 ) )
= A3 )
=> ( ( linorder_insort_key @ B @ A @ F3 @ A3 @ ( remove1 @ B @ A3 @ Xs2 ) )
= Xs2 ) ) ) ) ) ).
% insort_key_remove1
thf(fact_6241_sorted__same,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [G3: ( list @ A ) > A,Xs2: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( filter2 @ A
@ ^ [X4: A] :
( X4
= ( G3 @ Xs2 ) )
@ Xs2 ) ) ) ).
% sorted_same
thf(fact_6242_sorted0,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% sorted0
thf(fact_6243_sorted__wrt__mono__rel,axiom,
! [A: $tType,Xs2: list @ A,P: A > A > $o,Q: A > A > $o] :
( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ( member @ A @ Y4 @ ( set2 @ A @ Xs2 ) )
=> ( ( P @ X5 @ Y4 )
=> ( Q @ X5 @ Y4 ) ) ) )
=> ( ( sorted_wrt @ A @ P @ Xs2 )
=> ( sorted_wrt @ A @ Q @ Xs2 ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_6244_strict__sorted__equal,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs2 )
=> ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys )
=> ( ( ( set2 @ A @ Ys )
= ( set2 @ A @ Xs2 ) )
=> ( Ys = Xs2 ) ) ) ) ) ).
% strict_sorted_equal
thf(fact_6245_sorted__wrt__append,axiom,
! [A: $tType,P: A > A > $o,Xs2: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ P @ ( append @ A @ Xs2 @ Ys ) )
= ( ( sorted_wrt @ A @ P @ Xs2 )
& ( sorted_wrt @ A @ P @ Ys )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Ys ) )
=> ( P @ X4 @ Y3 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_6246_sorted__drop,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,N: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( drop @ A @ N @ Xs2 ) ) ) ) ).
% sorted_drop
thf(fact_6247_sorted__take,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,N: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( take @ A @ N @ Xs2 ) ) ) ) ).
% sorted_take
thf(fact_6248_sorted__nths,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,I5: set @ nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nths @ A @ Xs2 @ I5 ) ) ) ) ).
% sorted_nths
thf(fact_6249_sorted__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups @ A @ Xs2 ) ) ) ) ).
% sorted_remdups
thf(fact_6250_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linord4507533701916653071of_set @ A @ A6 ) ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_6251_sorted__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,A3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remove1 @ A @ A3 @ Xs2 ) ) ) ) ).
% sorted_remove1
thf(fact_6252_strict__sorted__imp__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 ) ) ) ).
% strict_sorted_imp_sorted
thf(fact_6253_sorted__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ Xs2 ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 ) ) ) ).
% sorted_insort
thf(fact_6254_sorted2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A,Zs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Zs2 ) ) )
= ( ( ord_less_eq @ A @ X3 @ Y )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y @ Zs2 ) ) ) ) ) ).
% sorted2
thf(fact_6255_sorted__replicate,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N: nat,X3: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( replicate @ A @ N @ X3 ) ) ) ).
% sorted_replicate
thf(fact_6256_sorted__upt,axiom,
! [M2: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M2 @ N ) ) ).
% sorted_upt
thf(fact_6257_sorted__map,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs2: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs2 ) )
= ( sorted_wrt @ B
@ ^ [X4: B,Y3: B] : ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) )
@ Xs2 ) ) ) ).
% sorted_map
thf(fact_6258_sorted1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% sorted1
thf(fact_6259_sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X3 @ Ys ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X3 @ X4 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys ) ) ) ) ).
% sorted_simps(2)
thf(fact_6260_strict__sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X3 @ Ys ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( ord_less @ A @ X3 @ X4 ) )
& ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys ) ) ) ) ).
% strict_sorted_simps(2)
thf(fact_6261_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_6262_sorted__append,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs2 @ Ys ) )
= ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ) ) ) ).
% sorted_append
thf(fact_6263_sorted__wrt_Oelims_I3_J,axiom,
! [A: $tType,X3: A > A > $o,Xa2: list @ A] :
( ~ ( sorted_wrt @ A @ X3 @ Xa2 )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ Ys4 ) )
=> ( ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Ys4 ) )
=> ( X3 @ X5 @ Xa3 ) )
& ( sorted_wrt @ A @ X3 @ Ys4 ) ) ) ) ).
% sorted_wrt.elims(3)
thf(fact_6264_sorted__wrt_Osimps_I2_J,axiom,
! [A: $tType,P: A > A > $o,X3: A,Ys: list @ A] :
( ( sorted_wrt @ A @ P @ ( cons @ A @ X3 @ Ys ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( P @ X3 @ X4 ) )
& ( sorted_wrt @ A @ P @ Ys ) ) ) ).
% sorted_wrt.simps(2)
thf(fact_6265_sorted__wrt__nth__less,axiom,
! [A: $tType,P: A > A > $o,Xs2: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ P @ Xs2 )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ I ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_6266_sorted__wrt__iff__nth__less,axiom,
! [A: $tType] :
( ( sorted_wrt @ A )
= ( ^ [P4: A > A > $o,Xs: list @ A] :
! [I4: nat,J3: nat] :
( ( ord_less @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P4 @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_6267_sorted__distinct__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ( distinct @ A @ Xs2 )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
=> ( ( distinct @ A @ Ys )
=> ( ( ( set2 @ A @ Xs2 )
= ( set2 @ A @ Ys ) )
=> ( Xs2 = Ys ) ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_6268_sorted__wrt01,axiom,
! [A: $tType,Xs2: list @ A,P: A > A > $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ P @ Xs2 ) ) ).
% sorted_wrt01
thf(fact_6269_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs2: list @ B,P: B > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs2 ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P @ Xs2 ) ) ) ) ) ).
% sorted_filter
thf(fact_6270_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Xs2: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs2 ) ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs2 ) ) ) ) ).
% sorted_insort_key
thf(fact_6271_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs2: list @ B,X3: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs2 ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( remove1 @ B @ X3 @ Xs2 ) ) ) ) ) ).
% sorted_map_remove1
thf(fact_6272_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,G3: ( list @ B ) > A,Xs2: list @ B] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( map @ B @ A @ F3
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F3 @ X4 )
= ( G3 @ Xs2 ) )
@ Xs2 ) ) ) ) ).
% sorted_map_same
thf(fact_6273_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
= ( ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I4 ) @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_6274_sorted01,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 ) ) ) ).
% sorted01
thf(fact_6275_sorted__wrt_Oelims_I1_J,axiom,
! [A: $tType,X3: A > A > $o,Xa2: list @ A,Y: $o] :
( ( ( sorted_wrt @ A @ X3 @ Xa2 )
= Y )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ~ Y )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ Ys4 ) )
=> ( Y
= ( ~ ( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Ys4 ) )
=> ( X3 @ X5 @ Y3 ) )
& ( sorted_wrt @ A @ X3 @ Ys4 ) ) ) ) ) ) ) ).
% sorted_wrt.elims(1)
thf(fact_6276_sorted__wrt_Oelims_I2_J,axiom,
! [A: $tType,X3: A > A > $o,Xa2: list @ A] :
( ( sorted_wrt @ A @ X3 @ Xa2 )
=> ( ( Xa2
!= ( nil @ A ) )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ Ys4 ) )
=> ~ ( ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) )
=> ( X3 @ X5 @ Xa ) )
& ( sorted_wrt @ A @ X3 @ Ys4 ) ) ) ) ) ).
% sorted_wrt.elims(2)
thf(fact_6277_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ? [X5: list @ A] :
( ( ( set2 @ A @ X5 )
= A6 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X5 )
& ( distinct @ A @ X5 )
& ! [Y6: list @ A] :
( ( ( ( set2 @ A @ Y6 )
= A6 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y6 )
& ( distinct @ A @ Y6 ) )
=> ( Y6 = X5 ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_6278_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ( distinct @ A @ Xs2 )
=> ( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs2 ) )
= Xs2 ) ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_6279_filter__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs2: list @ B,P: B > $o,X3: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs2 ) )
=> ( ( P @ X3 )
=> ( ( filter2 @ B @ P @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs2 ) )
= ( linorder_insort_key @ B @ A @ F3 @ X3 @ ( filter2 @ B @ P @ Xs2 ) ) ) ) ) ) ).
% filter_insort
thf(fact_6280_insort__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,Xs2: list @ A] :
( ( member @ A @ A3 @ ( set2 @ A @ Xs2 ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ A3
@ ( remove1 @ A @ A3 @ Xs2 ) )
= Xs2 ) ) ) ) ).
% insort_remove1
thf(fact_6281_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
= ( ! [I4: nat] :
( ( ord_less @ nat @ ( suc @ I4 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I4 ) @ ( nth @ A @ Xs2 @ ( suc @ I4 ) ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_6282_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I4 ) @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_6283_sorted__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_6284_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ~ ! [L4: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L4 )
=> ( ( ( set2 @ A @ L4 )
= A6 )
=> ( ( size_size @ ( list @ A ) @ L4 )
!= ( finite_card @ A @ A6 ) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_6285_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_6286_sorted__enumerate,axiom,
! [A: $tType,N: nat,Xs2: list @ A] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs2 ) ) ) ).
% sorted_enumerate
thf(fact_6287_map__sorted__distinct__set__unique,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs2: list @ B,Ys: list @ B] :
( ( inj_on @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs2 ) @ ( set2 @ B @ Ys ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs2 ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F3 @ Xs2 ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Ys ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F3 @ Ys ) )
=> ( ( ( set2 @ B @ Xs2 )
= ( set2 @ B @ Ys ) )
=> ( Xs2 = Ys ) ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_6288_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,L: list @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
& ( ( set2 @ A @ L )
= A6 )
& ( ( size_size @ ( list @ A ) @ L )
= ( finite_card @ A @ A6 ) ) )
= ( ( linord4507533701916653071of_set @ A @ A6 )
= L ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_6289_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,A3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ X5 @ A3 ) )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ A3
@ Xs2 )
= ( append @ A @ Xs2 @ ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_insort_is_snoc
thf(fact_6290_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),I: nat,J: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) ) )
=> ( ( ord_less @ nat @ J
@ ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs2 ) ) )
=> ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs2 ) @ I ) @ J )
= ( nth @ A @ ( nth @ ( list @ A ) @ Xs2 @ J ) @ I ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_6291_transpose__column,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) )
=> ( ( 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 @ Xs2 ) ) )
= ( nth @ ( list @ A ) @ Xs2 @ I ) ) ) ) ).
% transpose_column
thf(fact_6292_set__rev,axiom,
! [A: $tType,Xs2: list @ A] :
( ( set2 @ A @ ( rev @ A @ Xs2 ) )
= ( set2 @ A @ Xs2 ) ) ).
% set_rev
thf(fact_6293_length__rev,axiom,
! [A: $tType,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs2 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_rev
thf(fact_6294_zip__rev,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( zip @ A @ B @ ( rev @ A @ Xs2 ) @ ( rev @ B @ Ys ) )
= ( rev @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) ) ) ).
% zip_rev
thf(fact_6295_drop__rev,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( drop @ A @ N @ ( rev @ A @ Xs2 ) )
= ( rev @ A @ ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) @ Xs2 ) ) ) ).
% drop_rev
thf(fact_6296_rev__drop,axiom,
! [A: $tType,I: nat,Xs2: list @ A] :
( ( rev @ A @ ( drop @ A @ I @ Xs2 ) )
= ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ I ) @ ( rev @ A @ Xs2 ) ) ) ).
% rev_drop
thf(fact_6297_rev__take,axiom,
! [A: $tType,I: nat,Xs2: list @ A] :
( ( rev @ A @ ( take @ A @ I @ Xs2 ) )
= ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ I ) @ ( rev @ A @ Xs2 ) ) ) ).
% rev_take
thf(fact_6298_take__rev,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( take @ A @ N @ ( rev @ A @ Xs2 ) )
= ( rev @ A @ ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) @ Xs2 ) ) ) ).
% take_rev
thf(fact_6299_rotate__rev,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( rotate @ A @ N @ ( rev @ A @ Xs2 ) )
= ( rev @ A @ ( rotate @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) @ Xs2 ) ) ) ).
% rotate_rev
thf(fact_6300_rev__nth,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( rev @ A @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_6301_rev__update,axiom,
! [A: $tType,K2: nat,Xs2: list @ A,Y: A] :
( ( ord_less @ nat @ K2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( rev @ A @ ( list_update @ A @ Xs2 @ K2 @ Y ) )
= ( list_update @ A @ ( rev @ A @ Xs2 ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ K2 ) @ ( one_one @ nat ) ) @ Y ) ) ) ).
% rev_update
thf(fact_6302_sorted__transpose,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) ) ) ) ).
% sorted_transpose
thf(fact_6303_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs2 ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ ( suc @ I4 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ ( suc @ I4 ) ) @ ( nth @ A @ Xs2 @ I4 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_6304_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I ) ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_6305_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs2 ) )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J3 ) @ ( nth @ A @ Xs2 @ I4 ) ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_6306_foldr__max__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,Y: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs2 ) )
=> ( ( ( Xs2
= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs2 @ Y )
= Y ) )
& ( ( Xs2
!= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs2 @ Y )
= ( ord_max @ A @ ( nth @ A @ Xs2 @ ( zero_zero @ nat ) ) @ Y ) ) ) ) ) ) ).
% foldr_max_sorted
thf(fact_6307_length__transpose__sorted,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( ( Xs2
= ( nil @ ( list @ A ) ) )
=> ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) )
= ( zero_zero @ nat ) ) )
& ( ( Xs2
!= ( nil @ ( list @ A ) ) )
=> ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs2 @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% length_transpose_sorted
thf(fact_6308_transpose__column__length,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) )
=> ( ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs2 ) ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs2 @ I ) ) ) ) ) ).
% transpose_column_length
thf(fact_6309_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A6: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ S3 )
=> ( ( finite_finite2 @ B @ A6 )
=> ~ ! [L4: list @ B] :
( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ L4 ) )
=> ( ( ( set2 @ B @ L4 )
= A6 )
=> ( ( size_size @ ( list @ B ) @ L4 )
!= ( finite_card @ B @ A6 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_6310_transpose__transpose,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( transpose @ A @ ( transpose @ A @ Xs2 ) )
= ( takeWhile @ ( list @ A )
@ ^ [X4: list @ A] :
( X4
!= ( nil @ A ) )
@ Xs2 ) ) ) ).
% transpose_transpose
thf(fact_6311_length__concat__rev,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( size_size @ ( list @ A ) @ ( concat @ A @ ( rev @ ( list @ A ) @ Xs2 ) ) )
= ( size_size @ ( list @ A ) @ ( concat @ A @ Xs2 ) ) ) ).
% length_concat_rev
thf(fact_6312_takeWhile__eq__all__conv,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( ( takeWhile @ A @ P @ Xs2 )
= Xs2 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X4 ) ) ) ) ).
% takeWhile_eq_all_conv
thf(fact_6313_takeWhile__append2,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o,Ys: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X5 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs2 @ Ys ) )
= ( append @ A @ Xs2 @ ( takeWhile @ A @ P @ Ys ) ) ) ) ).
% takeWhile_append2
thf(fact_6314_takeWhile__append1,axiom,
! [A: $tType,X3: A,Xs2: list @ A,P: A > $o,Ys: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ~ ( P @ X3 )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs2 @ Ys ) )
= ( takeWhile @ A @ P @ Xs2 ) ) ) ) ).
% takeWhile_append1
thf(fact_6315_set__takeWhileD,axiom,
! [A: $tType,X3: A,P: A > $o,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( takeWhile @ A @ P @ Xs2 ) ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
& ( P @ X3 ) ) ) ).
% set_takeWhileD
thf(fact_6316_takeWhile__cong,axiom,
! [A: $tType,L: list @ A,K2: list @ A,P: A > $o,Q: A > $o] :
( ( L = K2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ L ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( takeWhile @ A @ P @ L )
= ( takeWhile @ A @ Q @ K2 ) ) ) ) ).
% takeWhile_cong
thf(fact_6317_takeWhile__eq__take,axiom,
! [A: $tType] :
( ( takeWhile @ A )
= ( ^ [P4: A > $o,Xs: list @ A] : ( take @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P4 @ Xs ) ) @ Xs ) ) ) ).
% takeWhile_eq_take
thf(fact_6318_length__takeWhile__le,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_takeWhile_le
thf(fact_6319_sorted__takeWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( takeWhile @ A @ P @ Xs2 ) ) ) ) ).
% sorted_takeWhile
thf(fact_6320_takeWhile__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs2: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs2 ) ) )
=> ( ( nth @ A @ ( takeWhile @ A @ P @ Xs2 ) @ J )
= ( nth @ A @ Xs2 @ J ) ) ) ).
% takeWhile_nth
thf(fact_6321_nth__length__takeWhile,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ~ ( P @ ( nth @ A @ Xs2 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs2 ) ) ) ) ) ).
% nth_length_takeWhile
thf(fact_6322_takeWhile__append,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o,Ys: list @ A] :
( ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X5 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs2 @ Ys ) )
= ( append @ A @ Xs2 @ ( takeWhile @ A @ P @ Ys ) ) ) )
& ( ~ ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( P @ X ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs2 @ Ys ) )
= ( takeWhile @ A @ P @ Xs2 ) ) ) ) ).
% takeWhile_append
thf(fact_6323_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs2: list @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( P @ ( nth @ A @ Xs2 @ I3 ) ) )
=> ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs2 ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_6324_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N: nat,Xs2: list @ A,P: A > $o] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ I3 ) ) ) )
=> ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ~ ( P @ ( nth @ A @ Xs2 @ N ) ) )
=> ( ( takeWhile @ A @ P @ Xs2 )
= ( take @ A @ N @ Xs2 ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_6325_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( folding_insort_key @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ ( set @ A ) )
@ ^ [X4: A] : X4 ) ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_6326_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs2: list @ B,T2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F3 @ Xs2 ) ) )
=> ( ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ T2 @ ( F3 @ X4 ) )
@ Xs2 )
= ( takeWhile @ B
@ ^ [X4: B] : ( ord_less @ A @ T2 @ ( F3 @ X4 ) )
@ Xs2 ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_6327_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A6: set @ B,L: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ S3 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ L ) )
& ( ( set2 @ B @ L )
= A6 )
& ( ( size_size @ ( list @ B ) @ L )
= ( finite_card @ B @ A6 ) ) )
= ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ A6 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_6328_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,X3: B,A6: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ X3 @ A6 ) @ S3 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert2 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( remove1 @ B @ X3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ A6 ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
thf(fact_6329_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A6: set @ B,B5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ S3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ S3 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ A6 )
= ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ B5 ) )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( A6 = B5 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_inject
thf(fact_6330_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ ( bot_bot @ ( set @ B ) ) )
= ( nil @ B ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_empty
thf(fact_6331_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A6: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ S3 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ( set2 @ B @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ A6 ) )
= A6 ) ) ) ) ).
% folding_insort_key.set_sorted_key_list_of_set
thf(fact_6332_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A6: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ S3 )
=> ( ( size_size @ ( list @ B ) @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ A6 ) )
= ( finite_card @ B @ A6 ) ) ) ) ).
% folding_insort_key.length_sorted_key_list_of_set
thf(fact_6333_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A6: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ S3 )
=> ( distinct @ A @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ A6 ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_6334_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A6: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ S3 )
=> ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ A6 ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_6335_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A6: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ S3 )
=> ( sorted_wrt @ A @ Less_eq2 @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ A6 ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_6336_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A6: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ S3 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ A6 )
= ( nil @ B ) )
= ( A6
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
thf(fact_6337_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,Xs2: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( set2 @ B @ Xs2 ) @ S3 )
=> ( ( sorted_wrt @ A @ Less_eq2 @ ( map @ B @ A @ F3 @ Xs2 ) )
=> ( ( distinct @ B @ Xs2 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ ( set2 @ B @ Xs2 ) )
= Xs2 ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_6338_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,X3: B,A6: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ X3 @ A6 ) @ S3 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ ( insert2 @ B @ X3 @ A6 ) )
= ( insort_key @ A @ B @ Less_eq2 @ F3 @ X3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert2 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
thf(fact_6339_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,X3: B,A6: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ X3 @ A6 ) @ S3 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ~ ( member @ B @ X3 @ A6 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ ( insert2 @ B @ X3 @ A6 ) )
= ( insort_key @ A @ B @ Less_eq2 @ F3 @ X3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F3 @ A6 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert
thf(fact_6340_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
& ( P @ X ) )
=> ( ( find @ A @ P @ Xs2 )
= ( some @ A
@ ( lattic643756798350308766er_Min @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
& ( P @ X4 ) ) ) ) ) ) ) ) ) ).
% sorted_find_Min
thf(fact_6341_total__on__singleton,axiom,
! [A: $tType,X3: A] : ( total_on @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% total_on_singleton
thf(fact_6342_total__on__diff__Id,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ A6 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) )
= ( total_on @ A @ A6 @ R2 ) ) ).
% total_on_diff_Id
thf(fact_6343_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_6344_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ A7 )
=> ( ( X4 != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_6345_total__onI,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ A6 )
=> ( ( member @ A @ Y4 @ A6 )
=> ( ( X5 != Y4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ R2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X5 ) @ R2 ) ) ) ) )
=> ( total_on @ A @ A6 @ R2 ) ) ).
% total_onI
thf(fact_6346_find__cong,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] :
( ( Xs2 = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( find @ A @ P @ Xs2 )
= ( find @ A @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_6347_find_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X3: A,Xs2: list @ A] :
( ( ( P @ X3 )
=> ( ( find @ A @ P @ ( cons @ A @ X3 @ Xs2 ) )
= ( some @ A @ X3 ) ) )
& ( ~ ( P @ X3 )
=> ( ( find @ A @ P @ ( cons @ A @ X3 @ Xs2 ) )
= ( find @ A @ P @ Xs2 ) ) ) ) ).
% find.simps(2)
thf(fact_6348_find_Osimps_I1_J,axiom,
! [A: $tType,Uu2: A > $o] :
( ( find @ A @ Uu2 @ ( nil @ A ) )
= ( none @ A ) ) ).
% find.simps(1)
thf(fact_6349_find__None__iff,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( ( find @ A @ P @ Xs2 )
= ( none @ A ) )
= ( ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
& ( P @ X4 ) ) ) ) ).
% find_None_iff
thf(fact_6350_find__None__iff2,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( ( none @ A )
= ( find @ A @ P @ Xs2 ) )
= ( ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
& ( P @ X4 ) ) ) ) ).
% find_None_iff2
thf(fact_6351_find__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A,X3: A] :
( ( ( find @ A @ P @ Xs2 )
= ( some @ A @ X3 ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P @ ( nth @ A @ Xs2 @ I4 ) )
& ( X3
= ( nth @ A @ Xs2 @ I4 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I4 )
=> ~ ( P @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_6352_find__Some__iff2,axiom,
! [A: $tType,X3: A,P: A > $o,Xs2: list @ A] :
( ( ( some @ A @ X3 )
= ( find @ A @ P @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P @ ( nth @ A @ Xs2 @ I4 ) )
& ( X3
= ( nth @ A @ Xs2 @ I4 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I4 )
=> ~ ( P @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_6353_remdups__adj__singleton__iff,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs2 ) )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( Xs2
!= ( nil @ A ) )
& ( Xs2
= ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( hd @ A @ Xs2 ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_6354_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,Y10: set @ A] :
( ( finite_finite2 @ A @ X8 )
& ( finite_finite2 @ A @ Y10 )
& ( Y10
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ Y10 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R6 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_6355_bex__empty,axiom,
! [A: $tType,P: A > $o] :
~ ? [X: A] :
( ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) )
& ( P @ X ) ) ).
% bex_empty
thf(fact_6356_remdups__adj__set,axiom,
! [A: $tType,Xs2: list @ A] :
( ( set2 @ A @ ( remdups_adj @ A @ Xs2 ) )
= ( set2 @ A @ Xs2 ) ) ).
% remdups_adj_set
thf(fact_6357_bex__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) )
& ( P @ X4 ) ) )
= ( ? [X8: A] : ( P @ X8 ) ) ) ).
% bex_UNIV
thf(fact_6358_remdups__adj__length,axiom,
! [A: $tType,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% remdups_adj_length
thf(fact_6359_Bex__def,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A7: set @ A,P4: A > $o] :
? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ( P4 @ X4 ) ) ) ) ).
% Bex_def
thf(fact_6360_image__def,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ A @ B )
= ( ^ [F4: A > B,A7: set @ A] :
( collect @ B
@ ^ [Y3: B] :
? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ( Y3
= ( F4 @ X4 ) ) ) ) ) ) ).
% image_def
thf(fact_6361_sorted__remdups__adj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups_adj @ A @ Xs2 ) ) ) ) ).
% sorted_remdups_adj
thf(fact_6362_max__extp_Ocases,axiom,
! [A: $tType,R: A > A > $o,A1: set @ A,A22: set @ A] :
( ( max_extp @ A @ R @ A1 @ A22 )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ~ ! [X: A] :
( ( member @ A @ X @ A1 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A22 )
& ( R @ X @ Xa3 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_6363_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 ) ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A12 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A23 )
& ( R6 @ X4 @ Y3 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_6364_max__extp_Omax__extI,axiom,
! [A: $tType,X6: set @ A,Y8: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ X6 )
=> ( ( finite_finite2 @ A @ Y8 )
=> ( ( Y8
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Y8 )
& ( R @ X5 @ Xa ) ) )
=> ( max_extp @ A @ R @ X6 @ Y8 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_6365_remdups__adj__adjacent,axiom,
! [A: $tType,I: nat,Xs2: list @ A] :
( ( ord_less @ nat @ ( suc @ I ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs2 ) ) )
=> ( ( nth @ A @ ( remdups_adj @ A @ Xs2 ) @ I )
!= ( nth @ A @ ( remdups_adj @ A @ Xs2 ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_6366_remdups__adj__singleton,axiom,
! [A: $tType,Xs2: list @ A,X3: A] :
( ( ( remdups_adj @ A @ Xs2 )
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( Xs2
= ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs2 ) @ X3 ) ) ) ).
% remdups_adj_singleton
thf(fact_6367_remdups__adj__length__ge1,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs2 ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_6368_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,Y10: set @ A] :
( ( Uu3
= ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X8 @ Y10 ) )
& ( X8
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ Y10 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ X8 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_6369_map__project__def,axiom,
! [B: $tType,A: $tType] :
( ( map_project @ A @ B )
= ( ^ [F4: A > ( option @ B ),A7: set @ A] :
( collect @ B
@ ^ [B8: B] :
? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ( ( F4 @ X4 )
= ( some @ B @ B8 ) ) ) ) ) ) ).
% map_project_def
thf(fact_6370_remdups__adj__altdef,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( ( remdups_adj @ A @ Xs2 )
= Ys )
= ( ? [F4: nat > nat] :
( ( order_mono @ nat @ nat @ F4 )
& ( ( image2 @ nat @ nat @ F4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I4 )
= ( nth @ A @ Ys @ ( F4 @ I4 ) ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( nth @ A @ Xs2 @ I4 )
= ( nth @ A @ Xs2 @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) )
= ( ( F4 @ I4 )
= ( F4 @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_6371_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_6372_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A3 ) ) ) ).
% mono_add
thf(fact_6373_mono__Suc,axiom,
order_mono @ nat @ nat @ suc ).
% mono_Suc
thf(fact_6374_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( ord_less @ A @ X3 @ Y ) ) ) ) ).
% mono_strict_invE
thf(fact_6375_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_6376_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_sup @ A )
& ( semilattice_sup @ B ) )
=> ! [F3: A > B,A6: A,B5: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F3 @ A6 ) @ ( F3 @ B5 ) ) @ ( F3 @ ( sup_sup @ A @ A6 @ B5 ) ) ) ) ) ).
% mono_sup
thf(fact_6377_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ).
% mono_invE
thf(fact_6378_incseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [F4: nat > A] :
! [N3: nat] : ( ord_less_eq @ A @ ( F4 @ N3 ) @ ( F4 @ ( suc @ N3 ) ) ) ) ) ) ).
% incseq_Suc_iff
thf(fact_6379_incseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X6: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
=> ( order_mono @ nat @ A @ X6 ) ) ) ).
% incseq_SucI
thf(fact_6380_incseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A6: nat > A,I: nat] :
( ( order_mono @ nat @ A @ A6 )
=> ( ord_less_eq @ A @ ( A6 @ I ) @ ( A6 @ ( suc @ I ) ) ) ) ) ).
% incseq_SucD
thf(fact_6381_monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ) ) ).
% monoD
thf(fact_6382_monoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ) ) ).
% monoE
thf(fact_6383_monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) )
=> ( order_mono @ A @ B @ F3 ) ) ) ).
% monoI
thf(fact_6384_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_mono @ A @ B )
= ( ^ [F4: A > B] :
! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( F4 @ Y3 ) ) ) ) ) ) ).
% mono_def
thf(fact_6385_incseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [X8: nat > A] :
! [M5: nat,N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) ) ) ) ) ).
% incseq_def
thf(fact_6386_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_6387_funpow__mono,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: A > A,A6: A,B5: A,N: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ A6 @ B5 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F3 @ A6 ) @ ( compow @ ( A > A ) @ N @ F3 @ B5 ) ) ) ) ) ).
% funpow_mono
thf(fact_6388_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_inf @ A )
& ( semilattice_inf @ B ) )
=> ! [F3: A > B,A6: A,B5: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( inf_inf @ A @ A6 @ B5 ) ) @ ( inf_inf @ B @ ( F3 @ A6 ) @ ( F3 @ B5 ) ) ) ) ) ).
% mono_inf
thf(fact_6389_cclfp__lowerbound,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F3: A > A,A6: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ A6 ) @ A6 )
=> ( ord_less_eq @ A @ ( order_532582986084564980_cclfp @ A @ F3 ) @ A6 ) ) ) ) ).
% cclfp_lowerbound
thf(fact_6390_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_6391_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_6392_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [F3: A > A,P2: A,K2: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ P2 @ ( F3 @ P2 ) )
=> ( ord_less_eq @ A @ P2 @ ( compow @ ( A > A ) @ K2 @ F3 @ ( top_top @ A ) ) ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_6393_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F3: A > A,P2: A,K2: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ P2 ) @ P2 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K2 @ F3 @ ( bot_bot @ A ) ) @ P2 ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_6394_funpow__mono2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: A > A,I: nat,J: nat,X3: A,Y: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ X3 @ ( F3 @ X3 ) )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ I @ F3 @ X3 ) @ ( compow @ ( A > A ) @ J @ F3 @ Y ) ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_6395_mono__Sup,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ A6 ) ) @ ( F3 @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% mono_Sup
thf(fact_6396_mono__SUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A6: C > A,I5: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F3 @ ( A6 @ X4 ) )
@ I5 ) )
@ ( F3 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A6 @ I5 ) ) ) ) ) ) ).
% mono_SUP
thf(fact_6397_mono__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ A6 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ A6 ) ) ) ) ) ).
% mono_Inf
thf(fact_6398_mono__INF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A6: C > A,I5: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A6 @ I5 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F3 @ ( A6 @ X4 ) )
@ I5 ) ) ) ) ) ).
% mono_INF
thf(fact_6399_incseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X6: nat > A,L5: A,N: nat] :
( ( order_mono @ nat @ A @ X6 )
=> ( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ ( X6 @ N ) @ L5 ) ) ) ) ).
% incseq_le
thf(fact_6400_funpow__increasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [M2: nat,N: nat,F3: A > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F3 @ ( top_top @ A ) ) @ ( compow @ ( A > A ) @ M2 @ F3 @ ( top_top @ A ) ) ) ) ) ) ).
% funpow_increasing
thf(fact_6401_funpow__decreasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [M2: nat,N: nat,F3: A > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M2 @ F3 @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N @ F3 @ ( bot_bot @ A ) ) ) ) ) ) ).
% funpow_decreasing
thf(fact_6402_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F3 @ ( lattic643756798349783984er_Max @ A @ A6 ) )
= ( lattic643756798349783984er_Max @ B @ ( image2 @ A @ B @ F3 @ A6 ) ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_6403_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F3 @ ( lattic643756798350308766er_Min @ A @ A6 ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F3 @ A6 ) ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_6404_mono__ge2__power__minus__self,axiom,
! [K2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ( order_mono @ nat @ nat
@ ^ [M5: nat] : ( minus_minus @ nat @ ( power_power @ nat @ K2 @ M5 ) @ M5 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_6405_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 )
=> ( ! [N2: nat] :
( ( ( F3 @ N2 )
= ( F3 @ ( suc @ N2 ) ) )
=> ( ( F3 @ ( suc @ N2 ) )
= ( F3 @ ( suc @ ( suc @ N2 ) ) ) ) )
=> ? [N8: nat] :
( ! [N9: nat] :
( ( ord_less_eq @ nat @ N9 @ N8 )
=> ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N8 )
=> ( ( ord_less @ nat @ M3 @ N9 )
=> ( ord_less @ A @ ( F3 @ M3 ) @ ( F3 @ N9 ) ) ) ) )
& ! [N9: nat] :
( ( ord_less_eq @ nat @ N8 @ N9 )
=> ( ( F3 @ N8 )
= ( F3 @ N9 ) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_6406_continuous__at__Sup__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,S3: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Sup_Sup @ A @ S3 ) @ ( set_ord_lessThan @ A @ ( complete_Sup_Sup @ A @ S3 ) ) ) @ F3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S3 )
=> ( ( F3 @ ( complete_Sup_Sup @ A @ S3 ) )
= ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ S3 ) ) ) ) ) ) ) ) ).
% continuous_at_Sup_mono
thf(fact_6407_continuous__at__Inf__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,S3: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Inf_Inf @ A @ S3 ) @ ( set_ord_greaterThan @ A @ ( complete_Inf_Inf @ A @ S3 ) ) ) @ F3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S3 )
=> ( ( F3 @ ( complete_Inf_Inf @ A @ S3 ) )
= ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ S3 ) ) ) ) ) ) ) ) ).
% continuous_at_Inf_mono
thf(fact_6408_bdd__below_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A6: set @ A,M7: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ A @ M7 @ X5 ) )
=> ( condit1013018076250108175_below @ A @ A6 ) ) ) ).
% bdd_below.I
thf(fact_6409_bdd__belowI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A6: set @ A,M2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ A @ M2 @ X5 ) )
=> ( condit1013018076250108175_below @ A @ A6 ) ) ) ).
% bdd_belowI
thf(fact_6410_bdd__above_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A6: set @ A,M7: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ A @ X5 @ M7 ) )
=> ( condit941137186595557371_above @ A @ A6 ) ) ) ).
% bdd_above.I
thf(fact_6411_bdd__below__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit1013018076250108175_below @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_below_empty
thf(fact_6412_bdd__above__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit941137186595557371_above @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_above_empty
thf(fact_6413_bdd__below__insert,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A3: A,A6: set @ A] :
( ( condit1013018076250108175_below @ A @ ( insert2 @ A @ A3 @ A6 ) )
= ( condit1013018076250108175_below @ A @ A6 ) ) ) ).
% bdd_below_insert
thf(fact_6414_bdd__above__insert,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A3: A,A6: set @ A] :
( ( condit941137186595557371_above @ A @ ( insert2 @ A @ A3 @ A6 ) )
= ( condit941137186595557371_above @ A @ A6 ) ) ) ).
% bdd_above_insert
thf(fact_6415_mono__Un,axiom,
! [B: $tType,A: $tType,F3: ( set @ A ) > ( set @ B ),A6: set @ A,B5: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F3 )
=> ( ord_less_eq @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F3 @ A6 ) @ ( F3 @ B5 ) ) @ ( F3 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) ) ) ) ).
% mono_Un
thf(fact_6416_mono__Int,axiom,
! [B: $tType,A: $tType,F3: ( set @ A ) > ( set @ B ),A6: set @ A,B5: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F3 )
=> ( ord_less_eq @ ( set @ B ) @ ( F3 @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) @ ( inf_inf @ ( set @ B ) @ ( F3 @ A6 ) @ ( F3 @ B5 ) ) ) ) ).
% mono_Int
thf(fact_6417_bdd__above__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B5: set @ A,A6: set @ A] :
( ( condit941137186595557371_above @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( condit941137186595557371_above @ A @ A6 ) ) ) ) ).
% bdd_above_mono
thf(fact_6418_bdd__below__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B5: set @ A,A6: set @ A] :
( ( condit1013018076250108175_below @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( condit1013018076250108175_below @ A @ A6 ) ) ) ) ).
% bdd_below_mono
thf(fact_6419_bdd__above_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit941137186595557371_above @ A )
= ( ^ [A7: set @ A] :
? [M9: A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ord_less_eq @ A @ X4 @ M9 ) ) ) ) ) ).
% bdd_above.unfold
thf(fact_6420_bdd__above_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A6: set @ A] :
( ( condit941137186595557371_above @ A @ A6 )
=> ~ ! [M8: A] :
~ ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ X @ M8 ) ) ) ) ).
% bdd_above.E
thf(fact_6421_bdd__below_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit1013018076250108175_below @ A )
= ( ^ [A7: set @ A] :
? [M9: A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ord_less_eq @ A @ M9 @ X4 ) ) ) ) ) ).
% bdd_below.unfold
thf(fact_6422_bdd__below_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A6: set @ A] :
( ( condit1013018076250108175_below @ A @ A6 )
=> ~ ! [M8: A] :
~ ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ M8 @ X ) ) ) ) ).
% bdd_below.E
thf(fact_6423_cSup__upper2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A,X6: set @ A,Y: A] :
( ( member @ A @ X3 @ X6 )
=> ( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( condit941137186595557371_above @ A @ X6 )
=> ( ord_less_eq @ A @ Y @ ( complete_Sup_Sup @ A @ X6 ) ) ) ) ) ) ).
% cSup_upper2
thf(fact_6424_cSup__upper,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A,X6: set @ A] :
( ( member @ A @ X3 @ X6 )
=> ( ( condit941137186595557371_above @ A @ X6 )
=> ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ X6 ) ) ) ) ) ).
% cSup_upper
thf(fact_6425_bdd__belowI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A6: set @ B,M2: A,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( ord_less_eq @ A @ M2 @ ( F3 @ X5 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ).
% bdd_belowI2
thf(fact_6426_bdd__below_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A6: set @ B,M7: A,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( ord_less_eq @ A @ M7 @ ( F3 @ X5 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ).
% bdd_below.I2
thf(fact_6427_bdd__above_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A6: set @ B,F3: B > A,M7: A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ M7 ) )
=> ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ).
% bdd_above.I2
thf(fact_6428_cInf__lower2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A,X6: set @ A,Y: A] :
( ( member @ A @ X3 @ X6 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( condit1013018076250108175_below @ A @ X6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X6 ) @ Y ) ) ) ) ) ).
% cInf_lower2
thf(fact_6429_cInf__lower,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A,X6: set @ A] :
( ( member @ A @ X3 @ X6 )
=> ( ( condit1013018076250108175_below @ A @ X6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X6 ) @ X3 ) ) ) ) ).
% cInf_lower
thf(fact_6430_cInf__le__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A6 )
=> ( ( condit1013018076250108175_below @ A @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_6431_cINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A6: set @ B,X3: B] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( F3 @ X3 ) ) ) ) ) ).
% cINF_lower
thf(fact_6432_cINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A6: set @ B,X3: B,U: A] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( member @ B @ X3 @ A6 )
=> ( ( ord_less_eq @ A @ ( F3 @ X3 ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ U ) ) ) ) ) ).
% cINF_lower2
thf(fact_6433_cInf__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B5: set @ A,A6: set @ A] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A6 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ B5 )
=> ? [X: A] :
( ( member @ A @ X @ A6 )
& ( ord_less_eq @ A @ X @ B4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B5 ) ) ) ) ) ) ).
% cInf_mono
thf(fact_6434_le__cInf__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S3: set @ A,A3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S3 )
=> ( ( ord_less_eq @ A @ A3 @ ( complete_Inf_Inf @ A @ S3 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ S3 )
=> ( ord_less_eq @ A @ A3 @ X4 ) ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_6435_cSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: B,A6: set @ B,F3: B > A] :
( ( member @ B @ X3 @ A6 )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ).
% cSUP_upper
thf(fact_6436_cSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A6: set @ B,X3: B,U: A] :
( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( member @ B @ X3 @ A6 )
=> ( ( ord_less_eq @ A @ U @ ( F3 @ X3 ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_6437_cInf__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,Y: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X6 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X6 ) @ Y )
= ( ? [X4: A] :
( ( member @ A @ X4 @ X6 )
& ( ord_less @ A @ X4 @ Y ) ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_6438_cSup__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B5: set @ A,A6: set @ A] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A6 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ B5 )
=> ? [X: A] :
( ( member @ A @ X @ A6 )
& ( ord_less_eq @ A @ B4 @ X ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B5 ) @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ).
% cSup_mono
thf(fact_6439_cSup__le__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S3: set @ A,A3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S3 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S3 ) @ A3 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ S3 )
=> ( ord_less_eq @ A @ X4 @ A3 ) ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_6440_less__cSup__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,Y: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X6 )
=> ( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X6 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ X6 )
& ( ord_less @ A @ Y @ X4 ) ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_6441_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B5: set @ B,F3: C > A,A6: set @ C,G3: B > A] :
( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ F3 @ A6 ) )
=> ( ! [M: B] :
( ( member @ B @ M @ B5 )
=> ? [X: C] :
( ( member @ C @ X @ A6 )
& ( ord_less_eq @ A @ ( F3 @ X ) @ ( G3 @ M ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F3 @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B5 ) ) ) ) ) ) ) ).
% cINF_mono
thf(fact_6442_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F3: B > A,U: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ord_less_eq @ A @ U @ ( F3 @ X4 ) ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_6443_cInf__superset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B5 ) @ ( complete_Inf_Inf @ A @ A6 ) ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_6444_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,G3: C > A,B5: set @ C,F3: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ G3 @ B5 ) )
=> ( ! [N2: B] :
( ( member @ B @ N2 @ A6 )
=> ? [X: C] :
( ( member @ C @ X @ B5 )
& ( ord_less_eq @ A @ ( F3 @ N2 ) @ ( G3 @ X ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G3 @ B5 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_6445_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F3: B > A,U: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ U )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ U ) ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_6446_cSup__subset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_6447_cInf__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,A3: A] :
( ( condit1013018076250108175_below @ A @ X6 )
=> ( ( ( X6
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ X6 ) )
= A3 ) )
& ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ X6 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ X6 ) ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_6448_cInf__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,A3: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X6 )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ X6 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ X6 ) ) ) ) ) ) ).
% cInf_insert
thf(fact_6449_cSup__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,A3: A] :
( ( condit941137186595557371_above @ A @ X6 )
=> ( ( ( X6
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ X6 ) )
= A3 ) )
& ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ X6 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ X6 ) ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_6450_cSup__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,A3: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X6 )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ X6 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ X6 ) ) ) ) ) ) ).
% cSup_insert
thf(fact_6451_cInf__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A6 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B5 )
=> ( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B5 ) ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_6452_cSup__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A6 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B5 )
=> ( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_6453_finite_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( set @ A ) > $o ) @ ( ( set @ A ) > $o )
@ ^ [P5: ( set @ A ) > $o,X4: set @ A] :
( ( X4
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,A8: A] :
( ( X4
= ( insert2 @ A @ A8 @ A7 ) )
& ( P5 @ A7 ) ) ) ) ).
% finite.mono
thf(fact_6454_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A6: set @ B,F3: B > A,A3: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ A3 )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( ord_less @ A @ ( F3 @ X4 ) @ A3 ) ) ) ) ) ) ) ).
% cINF_less_iff
thf(fact_6455_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A6: set @ B,F3: B > A,A3: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( ord_less @ A @ A3 @ ( F3 @ X4 ) ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_6456_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F3: B > A,G3: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G3 @ A6 ) )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ A6 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A8: B] : ( inf_inf @ A @ ( F3 @ A8 ) @ ( G3 @ A8 ) )
@ A6 ) ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_6457_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F3: B > A,G3: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G3 @ A6 ) )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ A6 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A8: B] : ( sup_sup @ A @ ( F3 @ A8 ) @ ( G3 @ A8 ) )
@ A6 ) ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_6458_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,G3: B > A,B5: set @ B,F3: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G3 @ B5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ B5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ B5 )
=> ( ord_less_eq @ A @ ( G3 @ X5 ) @ ( F3 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_6459_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,G3: B > A,B5: set @ B,F3: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G3 @ B5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ B5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_6460_less__eq__cInf__inter,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( condit1013018076250108175_below @ A @ A6 )
=> ( ( condit1013018076250108175_below @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A6 @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B5 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_6461_cINF__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F3: B > A,A3: B] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( insert2 @ B @ A3 @ A6 ) ) )
= ( inf_inf @ A @ ( F3 @ A3 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ) ).
% cINF_insert
thf(fact_6462_cSup__inter__less__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( condit941137186595557371_above @ A @ A6 )
=> ( ( condit941137186595557371_above @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A6 @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_6463_cSUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F3: B > A,A3: B] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( insert2 @ B @ A3 @ A6 ) ) )
= ( sup_sup @ A @ ( F3 @ A3 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_6464_cINF__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F3: B > A,B5: set @ B] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ B5 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A6 @ B5 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ B5 ) ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_6465_cSUP__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F3: B > A,B5: set @ B] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ B5 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A6 @ B5 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ B5 ) ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_6466_cInf__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S3: set @ A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S3 )
=> ( ( complete_Inf_Inf @ A @ S3 )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [X4: A] :
! [Y3: A] :
( ( member @ A @ Y3 @ S3 )
=> ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_6467_cSup__cInf,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S3: set @ A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S3 )
=> ( ( complete_Sup_Sup @ A @ S3 )
= ( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [X4: A] :
! [Y3: A] :
( ( member @ A @ Y3 @ S3 )
=> ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_6468_mono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A6: C > A,I5: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ A6 @ I5 ) )
=> ( ( I5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A6 @ I5 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F3 @ ( A6 @ X4 ) )
@ I5 ) ) ) ) ) ) ) ).
% mono_cINF
thf(fact_6469_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit1013018076250108175_below @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ A6 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ A6 ) ) ) ) ) ) ) ).
% mono_cInf
thf(fact_6470_cINF__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A6: set @ C,B5: C > ( set @ D ),F3: D > B] :
( ( A6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ A6 )
=> ( ( B5 @ X5 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit1013018076250108175_below @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X4: C] : ( image2 @ D @ B @ F3 @ ( B5 @ X4 ) )
@ A6 ) ) )
=> ( ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F3 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B5 @ A6 ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F3 @ ( B5 @ X4 ) ) )
@ A6 ) ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_6471_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A6: C > A,I5: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ A6 @ I5 ) )
=> ( ( I5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F3 @ ( A6 @ X4 ) )
@ I5 ) )
@ ( F3 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A6 @ I5 ) ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_6472_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit941137186595557371_above @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ A6 ) ) @ ( F3 @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ) ).
% mono_cSup
thf(fact_6473_cSUP__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A6: set @ C,B5: C > ( set @ D ),F3: D > B] :
( ( A6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ A6 )
=> ( ( B5 @ X5 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit941137186595557371_above @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X4: C] : ( image2 @ D @ B @ F3 @ ( B5 @ X4 ) )
@ A6 ) ) )
=> ( ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F3 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B5 @ A6 ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F3 @ ( B5 @ X4 ) ) )
@ A6 ) ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_6474_continuous__at__Inf__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,S3: set @ A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Inf_Inf @ A @ S3 ) @ ( set_ord_greaterThan @ A @ ( complete_Inf_Inf @ A @ S3 ) ) ) @ F3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S3 )
=> ( ( F3 @ ( complete_Inf_Inf @ A @ S3 ) )
= ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ S3 ) ) ) ) ) ) ) ) ).
% continuous_at_Inf_antimono
thf(fact_6475_continuous__at__Sup__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,S3: set @ A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Sup_Sup @ A @ S3 ) @ ( set_ord_lessThan @ A @ ( complete_Sup_Sup @ A @ S3 ) ) ) @ F3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S3 )
=> ( ( F3 @ ( complete_Sup_Sup @ A @ S3 ) )
= ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ S3 ) ) ) ) ) ) ) ) ).
% continuous_at_Sup_antimono
thf(fact_6476_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,X4: D] : ( Q @ I4 @ ( F3 @ X4 ) ) ) ) ) ).
% mono_compose
thf(fact_6477_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_6478_decseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X8: nat > A] :
! [M5: nat,N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) ) ) ) ) ).
% decseq_def
thf(fact_6479_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X3 ) ) ) ) ) ).
% antimonoD
thf(fact_6480_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X3 ) ) ) ) ) ).
% antimonoE
thf(fact_6481_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ Y4 ) @ ( F3 @ X5 ) ) )
=> ( order_antimono @ A @ B @ F3 ) ) ) ).
% antimonoI
thf(fact_6482_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F4: A > B] :
! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F4 @ Y3 ) @ ( F4 @ X4 ) ) ) ) ) ) ).
% antimono_def
thf(fact_6483_decseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A6: nat > A,I: nat] :
( ( order_antimono @ nat @ A @ A6 )
=> ( ord_less_eq @ A @ ( A6 @ ( suc @ I ) ) @ ( A6 @ I ) ) ) ) ).
% decseq_SucD
thf(fact_6484_decseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X6: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) )
=> ( order_antimono @ nat @ A @ X6 ) ) ) ).
% decseq_SucI
thf(fact_6485_decseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F4: nat > A] :
! [N3: nat] : ( ord_less_eq @ A @ ( F4 @ ( suc @ N3 ) ) @ ( F4 @ N3 ) ) ) ) ) ).
% decseq_Suc_iff
thf(fact_6486_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_6487_decseq__ge,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X6: nat > A,L5: A,N: nat] :
( ( order_antimono @ nat @ A @ X6 )
=> ( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ L5 @ ( X6 @ N ) ) ) ) ) ).
% decseq_ge
thf(fact_6488_iteratesp_Omono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F3: A > A] :
( order_mono @ ( A > $o ) @ ( A > $o )
@ ^ [P5: A > $o,X4: A] :
( ? [Y3: A] :
( ( X4
= ( F3 @ Y3 ) )
& ( P5 @ Y3 ) )
| ? [M9: set @ A] :
( ( X4
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ M9 )
=> ( P5 @ Y3 ) ) ) ) ) ) ).
% iteratesp.mono
thf(fact_6489_prod_Oset__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: B > A,Xs2: list @ B] :
( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( set2 @ B @ Xs2 ) )
= ( groups5270119922927024881d_list @ A @ ( map @ B @ A @ G3 @ ( remdups @ B @ Xs2 ) ) ) ) ) ).
% prod.set_conv_list
thf(fact_6490_chain__subset,axiom,
! [A: $tType,Ord: A > A > $o,A6: set @ A,B5: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( comple1602240252501008431_chain @ A @ Ord @ B5 ) ) ) ).
% chain_subset
thf(fact_6491_chain__empty,axiom,
! [A: $tType,Ord: A > A > $o] : ( comple1602240252501008431_chain @ A @ Ord @ ( bot_bot @ ( set @ A ) ) ) ).
% chain_empty
thf(fact_6492_ccpo__Sup__upper,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A6: set @ A,X3: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_6493_ccpo__Sup__least,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A6: set @ A,Z2: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ A @ X5 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ Z2 ) ) ) ) ).
% ccpo_Sup_least
thf(fact_6494_prod__list__zero__iff,axiom,
! [A: $tType] :
( ( ( semiring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [Xs2: list @ A] :
( ( ( groups5270119922927024881d_list @ A @ Xs2 )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ ( set2 @ A @ Xs2 ) ) ) ) ).
% prod_list_zero_iff
thf(fact_6495_chain__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X3: A] : ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% chain_singleton
thf(fact_6496_prod_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Xs2: list @ B,G3: B > A] :
( ( distinct @ B @ Xs2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( set2 @ B @ Xs2 ) )
= ( groups5270119922927024881d_list @ A @ ( map @ B @ A @ G3 @ Xs2 ) ) ) ) ) ).
% prod.distinct_set_conv_list
thf(fact_6497_in__chain__finite,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A6: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A6 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A6 ) @ A6 ) ) ) ) ) ).
% in_chain_finite
thf(fact_6498_prod__encode__prod__decode__aux,axiom,
! [K2: nat,M2: nat] :
( ( nat_prod_encode @ ( nat_prod_decode_aux @ K2 @ M2 ) )
= ( plus_plus @ nat @ ( nat_triangle @ K2 ) @ M2 ) ) ).
% prod_encode_prod_decode_aux
thf(fact_6499_prod__encode__eq,axiom,
! [X3: product_prod @ nat @ nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_encode @ X3 )
= ( nat_prod_encode @ Y ) )
= ( X3 = Y ) ) ).
% prod_encode_eq
thf(fact_6500_surj__prod__encode,axiom,
( ( image2 @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_prod_encode
thf(fact_6501_bij__prod__encode,axiom,
bij_betw @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% bij_prod_encode
thf(fact_6502_inj__prod__encode,axiom,
! [A6: set @ ( product_prod @ nat @ nat )] : ( inj_on @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ A6 ) ).
% inj_prod_encode
thf(fact_6503_le__prod__encode__2,axiom,
! [B2: nat,A3: nat] : ( ord_less_eq @ nat @ B2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% le_prod_encode_2
thf(fact_6504_le__prod__encode__1,axiom,
! [A3: nat,B2: nat] : ( ord_less_eq @ nat @ A3 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% le_prod_encode_1
thf(fact_6505_prod__encode__def,axiom,
( nat_prod_encode
= ( product_case_prod @ nat @ nat @ nat
@ ^ [M5: nat,N3: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M5 @ N3 ) ) @ M5 ) ) ) ).
% prod_encode_def
thf(fact_6506_list__encode_Oelims,axiom,
! [X3: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X3 )
= Y )
=> ( ( ( X3
= ( nil @ nat ) )
=> ( Y
!= ( zero_zero @ nat ) ) )
=> ~ ! [X5: nat,Xs3: list @ nat] :
( ( X3
= ( cons @ nat @ X5 @ Xs3 ) )
=> ( Y
!= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_6507_flat__lub__def,axiom,
! [A: $tType] :
( ( partial_flat_lub @ A )
= ( ^ [B8: A,A7: set @ A] :
( if @ A @ ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert2 @ A @ B8 @ ( bot_bot @ ( set @ A ) ) ) ) @ B8
@ ( the @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert2 @ A @ B8 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_6508_inj__list__encode,axiom,
! [A6: set @ ( list @ nat )] : ( inj_on @ ( list @ nat ) @ nat @ nat_list_encode @ A6 ) ).
% inj_list_encode
thf(fact_6509_list__encode__eq,axiom,
! [X3: list @ nat,Y: list @ nat] :
( ( ( nat_list_encode @ X3 )
= ( nat_list_encode @ Y ) )
= ( X3 = Y ) ) ).
% list_encode_eq
thf(fact_6510_bij__list__encode,axiom,
bij_betw @ ( list @ nat ) @ nat @ nat_list_encode @ ( top_top @ ( set @ ( list @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% bij_list_encode
thf(fact_6511_surj__list__encode,axiom,
( ( image2 @ ( list @ nat ) @ nat @ nat_list_encode @ ( top_top @ ( set @ ( list @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_list_encode
thf(fact_6512_list__encode_Osimps_I1_J,axiom,
( ( nat_list_encode @ ( nil @ nat ) )
= ( zero_zero @ nat ) ) ).
% list_encode.simps(1)
thf(fact_6513_list__encode_Osimps_I2_J,axiom,
! [X3: nat,Xs2: list @ nat] :
( ( nat_list_encode @ ( cons @ nat @ X3 @ Xs2 ) )
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).
% list_encode.simps(2)
thf(fact_6514_list__encode_Opelims,axiom,
! [X3: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X3 )
= Y )
=> ( ( accp @ ( list @ nat ) @ nat_list_encode_rel @ X3 )
=> ( ( ( X3
= ( nil @ nat ) )
=> ( ( Y
= ( zero_zero @ nat ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( nil @ nat ) ) ) )
=> ~ ! [X5: nat,Xs3: list @ nat] :
( ( X3
= ( cons @ nat @ X5 @ Xs3 ) )
=> ( ( Y
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X5 @ Xs3 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_6515_card__quotient__disjoint,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A6 )
=> ( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A] : ( equiv_quotient @ A @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
@ A6 )
=> ( ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A6 @ R2 ) )
= ( finite_card @ A @ A6 ) ) ) ) ).
% card_quotient_disjoint
thf(fact_6516_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_6517_quotient__is__empty,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ( equiv_quotient @ A @ A6 @ R2 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty
thf(fact_6518_quotient__is__empty2,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ( bot_bot @ ( set @ ( set @ A ) ) )
= ( equiv_quotient @ A @ A6 @ R2 ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty2
thf(fact_6519_quotient__diff1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A,A3: A] :
( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [A8: A] : ( equiv_quotient @ A @ ( insert2 @ A @ A8 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
@ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ( equiv_quotient @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ R2 )
= ( minus_minus @ ( set @ ( set @ A ) ) @ ( equiv_quotient @ A @ A6 @ R2 ) @ ( equiv_quotient @ A @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) @ R2 ) ) ) ) ) ).
% quotient_diff1
thf(fact_6520_INF__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,Xs2: list @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs2 ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F3 ) @ Xs2 @ ( top_top @ A ) ) ) ) ).
% INF_set_fold
thf(fact_6521_SUP__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,Xs2: list @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs2 ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F3 ) @ Xs2 @ ( bot_bot @ A ) ) ) ) ).
% SUP_set_fold
thf(fact_6522_fold__invariant,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Q: A > $o,P: B > $o,S: B,F3: A > B > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( Q @ X5 ) )
=> ( ( P @ S )
=> ( ! [X5: A,S2: B] :
( ( Q @ X5 )
=> ( ( P @ S2 )
=> ( P @ ( F3 @ X5 @ S2 ) ) ) )
=> ( P @ ( fold @ A @ B @ F3 @ Xs2 @ S ) ) ) ) ) ).
% fold_invariant
thf(fact_6523_List_Ofold__cong,axiom,
! [B: $tType,A: $tType,A3: A,B2: A,Xs2: list @ B,Ys: list @ B,F3: B > A > A,G3: B > A > A] :
( ( A3 = B2 )
=> ( ( Xs2 = Ys )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ Xs2 ) )
=> ( ( F3 @ X5 )
= ( G3 @ X5 ) ) )
=> ( ( fold @ B @ A @ F3 @ Xs2 @ A3 )
= ( fold @ B @ A @ G3 @ Ys @ B2 ) ) ) ) ) ).
% List.fold_cong
thf(fact_6524_fold__commute,axiom,
! [A: $tType,C: $tType,B: $tType,Xs2: list @ A,H: B > C,G3: A > B > B,F3: A > C > C] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ( comp @ B @ C @ B @ H @ ( G3 @ X5 ) )
= ( comp @ C @ C @ B @ ( F3 @ X5 ) @ H ) ) )
=> ( ( comp @ B @ C @ B @ H @ ( fold @ A @ B @ G3 @ Xs2 ) )
= ( comp @ C @ C @ B @ ( fold @ A @ C @ F3 @ Xs2 ) @ H ) ) ) ).
% fold_commute
thf(fact_6525_fold__commute__apply,axiom,
! [A: $tType,C: $tType,B: $tType,Xs2: list @ A,H: B > C,G3: A > B > B,F3: A > C > C,S: B] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ( comp @ B @ C @ B @ H @ ( G3 @ X5 ) )
= ( comp @ C @ C @ B @ ( F3 @ X5 ) @ H ) ) )
=> ( ( H @ ( fold @ A @ B @ G3 @ Xs2 @ S ) )
= ( fold @ A @ C @ F3 @ Xs2 @ ( H @ S ) ) ) ) ).
% fold_commute_apply
thf(fact_6526_union__set__fold,axiom,
! [A: $tType,Xs2: list @ A,A6: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A6 )
= ( fold @ A @ ( set @ A ) @ ( insert2 @ A ) @ Xs2 @ A6 ) ) ).
% union_set_fold
thf(fact_6527_fold__rev,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,F3: A > B > B] :
( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ( member @ A @ Y4 @ ( set2 @ A @ Xs2 ) )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y4 ) @ ( F3 @ X5 ) )
= ( comp @ B @ B @ B @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) ) ) )
=> ( ( fold @ A @ B @ F3 @ ( rev @ A @ Xs2 ) )
= ( fold @ A @ B @ F3 @ Xs2 ) ) ) ).
% fold_rev
thf(fact_6528_fold__plus__sum__list__rev,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs2: list @ A] :
( ( fold @ A @ A @ ( plus_plus @ A ) @ Xs2 )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( rev @ A @ Xs2 ) ) ) ) ) ).
% fold_plus_sum_list_rev
thf(fact_6529_fold__remove1__split,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,F3: A > B > B,X3: A] :
( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ( member @ A @ Y4 @ ( set2 @ A @ Xs2 ) )
=> ( ( comp @ B @ B @ B @ ( F3 @ X5 ) @ ( F3 @ Y4 ) )
= ( comp @ B @ B @ B @ ( F3 @ Y4 ) @ ( F3 @ X5 ) ) ) ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( fold @ A @ B @ F3 @ Xs2 )
= ( comp @ B @ B @ B @ ( fold @ A @ B @ F3 @ ( remove1 @ A @ X3 @ Xs2 ) ) @ ( F3 @ X3 ) ) ) ) ) ).
% fold_remove1_split
thf(fact_6530_foldr__fold,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,F3: A > B > B] :
( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ( member @ A @ Y4 @ ( set2 @ A @ Xs2 ) )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y4 ) @ ( F3 @ X5 ) )
= ( comp @ B @ B @ B @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) ) ) )
=> ( ( foldr @ A @ B @ F3 @ Xs2 )
= ( fold @ A @ B @ F3 @ Xs2 ) ) ) ).
% foldr_fold
thf(fact_6531_Sup__set__fold,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Xs2: list @ A] :
( ( complete_Sup_Sup @ A @ ( set2 @ A @ Xs2 ) )
= ( fold @ A @ A @ ( sup_sup @ A ) @ Xs2 @ ( bot_bot @ A ) ) ) ) ).
% Sup_set_fold
thf(fact_6532_Inf__set__fold,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Xs2: list @ A] :
( ( complete_Inf_Inf @ A @ ( set2 @ A @ Xs2 ) )
= ( fold @ A @ A @ ( inf_inf @ A ) @ Xs2 @ ( top_top @ A ) ) ) ) ).
% Inf_set_fold
thf(fact_6533_Gcd__set__eq__fold,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [Xs2: list @ A] :
( ( gcd_Gcd @ A @ ( set2 @ A @ Xs2 ) )
= ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs2 @ ( zero_zero @ A ) ) ) ) ).
% Gcd_set_eq_fold
thf(fact_6534_Inf__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Xs2: list @ A] :
( ( lattic7752659483105999362nf_fin @ A @ ( set2 @ A @ ( cons @ A @ X3 @ Xs2 ) ) )
= ( fold @ A @ A @ ( inf_inf @ A ) @ Xs2 @ X3 ) ) ) ).
% Inf_fin.set_eq_fold
thf(fact_6535_Sup__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Xs2: list @ A] :
( ( lattic5882676163264333800up_fin @ A @ ( set2 @ A @ ( cons @ A @ X3 @ Xs2 ) ) )
= ( fold @ A @ A @ ( sup_sup @ A ) @ Xs2 @ X3 ) ) ) ).
% Sup_fin.set_eq_fold
thf(fact_6536_Max_Oset__eq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Xs2: list @ A] :
( ( lattic643756798349783984er_Max @ A @ ( set2 @ A @ ( cons @ A @ X3 @ Xs2 ) ) )
= ( fold @ A @ A @ ( ord_max @ A ) @ Xs2 @ X3 ) ) ) ).
% Max.set_eq_fold
thf(fact_6537_Min_Oset__eq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Xs2: list @ A] :
( ( lattic643756798350308766er_Min @ A @ ( set2 @ A @ ( cons @ A @ X3 @ Xs2 ) ) )
= ( fold @ A @ A @ ( ord_min @ A ) @ Xs2 @ X3 ) ) ) ).
% Min.set_eq_fold
thf(fact_6538_comp__fun__idem__on_Ofold__set__fold,axiom,
! [A: $tType,B: $tType,S3: set @ A,F3: A > B > B,Xs2: list @ A,Y: B] :
( ( finite673082921795544331dem_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ S3 )
=> ( ( finite_fold @ A @ B @ F3 @ Y @ ( set2 @ A @ Xs2 ) )
= ( fold @ A @ B @ F3 @ Xs2 @ Y ) ) ) ) ).
% comp_fun_idem_on.fold_set_fold
thf(fact_6539_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
! [A: $tType,B: $tType,S3: set @ A,F3: A > B > B,Xs2: list @ A,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ S3 )
=> ( ( finite_fold @ A @ B @ F3 @ Y @ ( set2 @ A @ Xs2 ) )
= ( fold @ A @ B @ F3 @ ( remdups @ A @ Xs2 ) @ Y ) ) ) ) ).
% comp_fun_commute_on.fold_set_fold_remdups
thf(fact_6540_finite__sequence__to__countable__set,axiom,
! [A: $tType,X6: set @ A] :
( ( countable_countable @ A @ X6 )
=> ~ ! [F5: nat > ( set @ A )] :
( ! [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( F5 @ I2 ) @ X6 )
=> ( ! [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( F5 @ I2 ) @ ( F5 @ ( suc @ I2 ) ) )
=> ( ! [I2: nat] : ( finite_finite2 @ A @ ( F5 @ I2 ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ F5 @ ( top_top @ ( set @ nat ) ) ) )
!= X6 ) ) ) ) ) ).
% finite_sequence_to_countable_set
thf(fact_6541_and__not__num_Opelims,axiom,
! [X3: num,Xa2: num,Y: option @ num] :
( ( ( bit_and_not_num @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ X3 @ Xa2 ) )
=> ( ( ( X3 = one2 )
=> ( ( Xa2 = one2 )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa2
= ( bit0 @ N2 ) )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa2
= ( bit1 @ N2 ) )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ( ( Xa2 = one2 )
=> ( ( Y
= ( some @ num @ ( bit0 @ M ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M ) @ one2 ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit0 @ N2 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit1 @ N2 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M ) @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ( ( Xa2 = one2 )
=> ( ( Y
= ( some @ num @ ( bit0 @ M ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M ) @ one2 ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit0 @ N2 ) )
=> ( ( Y
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N10: num] : ( some @ num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ~ ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit1 @ N2 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
thf(fact_6542_countable__empty,axiom,
! [A: $tType] : ( countable_countable @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% countable_empty
thf(fact_6543_countable__insert,axiom,
! [A: $tType,A6: set @ A,A3: A] :
( ( countable_countable @ A @ A6 )
=> ( countable_countable @ A @ ( insert2 @ A @ A3 @ A6 ) ) ) ).
% countable_insert
thf(fact_6544_countable__insert__eq,axiom,
! [A: $tType,X3: A,A6: set @ A] :
( ( countable_countable @ A @ ( insert2 @ A @ X3 @ A6 ) )
= ( countable_countable @ A @ A6 ) ) ).
% countable_insert_eq
thf(fact_6545_ccSup__insert,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,A3: A] :
( ( countable_countable @ A @ A6 )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ A6 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% ccSup_insert
thf(fact_6546_ccInf__insert,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,A3: A] :
( ( countable_countable @ A @ A6 )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ A6 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ A6 ) ) ) ) ) ).
% ccInf_insert
thf(fact_6547_countable__Diff__eq,axiom,
! [A: $tType,A6: set @ A,X3: A] :
( ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( countable_countable @ A @ A6 ) ) ).
% countable_Diff_eq
thf(fact_6548_ccSUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,F3: B > A,A3: B] :
( ( countable_countable @ B @ A6 )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( insert2 @ B @ A3 @ A6 ) ) )
= ( sup_sup @ A @ ( F3 @ A3 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ).
% ccSUP_insert
thf(fact_6549_ccINF__insert,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,F3: B > A,A3: B] :
( ( countable_countable @ B @ A6 )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( insert2 @ B @ A3 @ A6 ) ) )
= ( inf_inf @ A @ ( F3 @ A3 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ).
% ccINF_insert
thf(fact_6550_countable__image__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,S3: set @ B] :
( ( countable_countable @ A @ ( image2 @ B @ A @ F3 @ S3 ) )
= ( ? [T9: set @ B] :
( ( countable_countable @ B @ T9 )
& ( ord_less_eq @ ( set @ B ) @ T9 @ S3 )
& ( ( image2 @ B @ A @ F3 @ S3 )
= ( image2 @ B @ A @ F3 @ T9 ) ) ) ) ) ).
% countable_image_eq
thf(fact_6551_countable__subset__image,axiom,
! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A6: set @ B] :
( ( ( countable_countable @ A @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ ( image2 @ B @ A @ F3 @ A6 ) ) )
= ( ? [A16: set @ B] :
( ( countable_countable @ B @ A16 )
& ( ord_less_eq @ ( set @ B ) @ A16 @ A6 )
& ( B5
= ( image2 @ B @ A @ F3 @ A16 ) ) ) ) ) ).
% countable_subset_image
thf(fact_6552_ex__countable__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,S3: set @ B,P: ( set @ A ) > $o] :
( ( ? [T9: set @ A] :
( ( countable_countable @ A @ T9 )
& ( ord_less_eq @ ( set @ A ) @ T9 @ ( image2 @ B @ A @ F3 @ S3 ) )
& ( P @ T9 ) ) )
= ( ? [T9: set @ B] :
( ( countable_countable @ B @ T9 )
& ( ord_less_eq @ ( set @ B ) @ T9 @ S3 )
& ( P @ ( image2 @ B @ A @ F3 @ T9 ) ) ) ) ) ).
% ex_countable_subset_image
thf(fact_6553_all__countable__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,S3: set @ B,P: ( set @ A ) > $o] :
( ( ! [T9: set @ A] :
( ( ( countable_countable @ A @ T9 )
& ( ord_less_eq @ ( set @ A ) @ T9 @ ( image2 @ B @ A @ F3 @ S3 ) ) )
=> ( P @ T9 ) ) )
= ( ! [T9: set @ B] :
( ( ( countable_countable @ B @ T9 )
& ( ord_less_eq @ ( set @ B ) @ T9 @ S3 ) )
=> ( P @ ( image2 @ B @ A @ F3 @ T9 ) ) ) ) ) ).
% all_countable_subset_image
thf(fact_6554_countable__subset,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( countable_countable @ A @ B5 )
=> ( countable_countable @ A @ A6 ) ) ) ).
% countable_subset
thf(fact_6555_ccSup__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B5: set @ A,A6: set @ A] :
( ( countable_countable @ A @ B5 )
=> ( ( countable_countable @ A @ A6 )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ? [X: A] :
( ( member @ A @ X @ B5 )
& ( ord_less_eq @ A @ A5 @ X ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ) ) ).
% ccSup_mono
thf(fact_6556_ccSup__least,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,Z2: A] :
( ( countable_countable @ A @ A6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ A @ X5 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ Z2 ) ) ) ) ).
% ccSup_least
thf(fact_6557_ccSup__upper,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,X3: A] :
( ( countable_countable @ A @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% ccSup_upper
thf(fact_6558_ccSup__le__iff,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,B2: A] :
( ( countable_countable @ A @ A6 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ B2 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ X4 @ B2 ) ) ) ) ) ) ).
% ccSup_le_iff
thf(fact_6559_ccSup__upper2,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,U: A,V2: A] :
( ( countable_countable @ A @ A6 )
=> ( ( member @ A @ U @ A6 )
=> ( ( ord_less_eq @ A @ V2 @ U )
=> ( ord_less_eq @ A @ V2 @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ).
% ccSup_upper2
thf(fact_6560_infinite__countable__subset_H,axiom,
! [A: $tType,X6: set @ A] :
( ~ ( finite_finite2 @ A @ X6 )
=> ? [C7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C7 @ X6 )
& ( countable_countable @ A @ C7 )
& ~ ( finite_finite2 @ A @ C7 ) ) ) ).
% infinite_countable_subset'
thf(fact_6561_countable__Collect__finite__subset,axiom,
! [A: $tType,T4: set @ A] :
( ( countable_countable @ A @ T4 )
=> ( countable_countable @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
& ( ord_less_eq @ ( set @ A ) @ A7 @ T4 ) ) ) ) ) ).
% countable_Collect_finite_subset
thf(fact_6562_ccInf__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B5: set @ A,A6: set @ A] :
( ( countable_countable @ A @ B5 )
=> ( ( countable_countable @ A @ A6 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ B5 )
=> ? [X: A] :
( ( member @ A @ X @ A6 )
& ( ord_less_eq @ A @ X @ B4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B5 ) ) ) ) ) ) ).
% ccInf_mono
thf(fact_6563_ccInf__lower,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,X3: A] :
( ( countable_countable @ A @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ X3 ) ) ) ) ).
% ccInf_lower
thf(fact_6564_ccInf__lower2,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,U: A,V2: A] :
( ( countable_countable @ A @ A6 )
=> ( ( member @ A @ U @ A6 )
=> ( ( ord_less_eq @ A @ U @ V2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ V2 ) ) ) ) ) ).
% ccInf_lower2
thf(fact_6565_le__ccInf__iff,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,B2: A] :
( ( countable_countable @ A @ A6 )
=> ( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A6 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ B2 @ X4 ) ) ) ) ) ) ).
% le_ccInf_iff
thf(fact_6566_ccInf__greatest,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,Z2: A] :
( ( countable_countable @ A @ A6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ A @ Z2 @ X5 ) )
=> ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ A6 ) ) ) ) ) ).
% ccInf_greatest
thf(fact_6567_ccSup__subset__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B5: set @ A,A6: set @ A] :
( ( countable_countable @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ) ).
% ccSup_subset_mono
thf(fact_6568_ccInf__superset__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( countable_countable @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B5 ) ) ) ) ) ).
% ccInf_superset_mono
thf(fact_6569_all__countable__subset__image__inj,axiom,
! [A: $tType,B: $tType,F3: B > A,S3: set @ B,P: ( set @ A ) > $o] :
( ( ! [T9: set @ A] :
( ( ( countable_countable @ A @ T9 )
& ( ord_less_eq @ ( set @ A ) @ T9 @ ( image2 @ B @ A @ F3 @ S3 ) ) )
=> ( P @ T9 ) ) )
= ( ! [T9: set @ B] :
( ( ( countable_countable @ B @ T9 )
& ( ord_less_eq @ ( set @ B ) @ T9 @ S3 )
& ( inj_on @ B @ A @ F3 @ T9 ) )
=> ( P @ ( image2 @ B @ A @ F3 @ T9 ) ) ) ) ) ).
% all_countable_subset_image_inj
thf(fact_6570_ex__countable__subset__image__inj,axiom,
! [A: $tType,B: $tType,F3: B > A,S3: set @ B,P: ( set @ A ) > $o] :
( ( ? [T9: set @ A] :
( ( countable_countable @ A @ T9 )
& ( ord_less_eq @ ( set @ A ) @ T9 @ ( image2 @ B @ A @ F3 @ S3 ) )
& ( P @ T9 ) ) )
= ( ? [T9: set @ B] :
( ( countable_countable @ B @ T9 )
& ( ord_less_eq @ ( set @ B ) @ T9 @ S3 )
& ( inj_on @ B @ A @ F3 @ T9 )
& ( P @ ( image2 @ B @ A @ F3 @ T9 ) ) ) ) ) ).
% ex_countable_subset_image_inj
thf(fact_6571_countable__image__eq__inj,axiom,
! [A: $tType,B: $tType,F3: B > A,S3: set @ B] :
( ( countable_countable @ A @ ( image2 @ B @ A @ F3 @ S3 ) )
= ( ? [T9: set @ B] :
( ( countable_countable @ B @ T9 )
& ( ord_less_eq @ ( set @ B ) @ T9 @ S3 )
& ( ( image2 @ B @ A @ F3 @ S3 )
= ( image2 @ B @ A @ F3 @ T9 ) )
& ( inj_on @ B @ A @ F3 @ T9 ) ) ) ) ).
% countable_image_eq_inj
thf(fact_6572_uncountable__def,axiom,
! [A: $tType,A6: set @ A] :
( ( ~ ( countable_countable @ A @ A6 ) )
= ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
& ~ ? [F4: nat > A] :
( ( image2 @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) )
= A6 ) ) ) ).
% uncountable_def
thf(fact_6573_ccSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,B5: set @ C,F3: B > A,G3: C > A] :
( ( countable_countable @ B @ A6 )
=> ( ( countable_countable @ C @ B5 )
=> ( ! [N2: B] :
( ( member @ B @ N2 @ A6 )
=> ? [X: C] :
( ( member @ C @ X @ B5 )
& ( ord_less_eq @ A @ ( F3 @ N2 ) @ ( G3 @ X ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G3 @ B5 ) ) ) ) ) ) ) ).
% ccSUP_mono
thf(fact_6574_ccSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,F3: B > A,U: A] :
( ( countable_countable @ B @ A6 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ U ) ) ) ) ).
% ccSUP_least
thf(fact_6575_ccSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,I: B,F3: B > A] :
( ( countable_countable @ B @ A6 )
=> ( ( member @ B @ I @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ I ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ).
% ccSUP_upper
thf(fact_6576_ccSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,F3: B > A,U: A] :
( ( countable_countable @ B @ A6 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ U )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ U ) ) ) ) ) ) ).
% ccSUP_le_iff
thf(fact_6577_ccSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,I: B,U: A,F3: B > A] :
( ( countable_countable @ B @ A6 )
=> ( ( member @ B @ I @ A6 )
=> ( ( ord_less_eq @ A @ U @ ( F3 @ I ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ) ).
% ccSUP_upper2
thf(fact_6578_ccINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,U: A,F3: B > A] :
( ( countable_countable @ B @ A6 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ord_less_eq @ A @ U @ ( F3 @ I3 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ).
% ccINF_greatest
thf(fact_6579_le__ccINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,U: A,F3: B > A] :
( ( countable_countable @ B @ A6 )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ord_less_eq @ A @ U @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% le_ccINF_iff
thf(fact_6580_ccINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,I: B,F3: B > A,U: A] :
( ( countable_countable @ B @ A6 )
=> ( ( member @ B @ I @ A6 )
=> ( ( ord_less_eq @ A @ ( F3 @ I ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ U ) ) ) ) ) ).
% ccINF_lower2
thf(fact_6581_ccINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,I: B,F3: B > A] :
( ( countable_countable @ B @ A6 )
=> ( ( member @ B @ I @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( F3 @ I ) ) ) ) ) ).
% ccINF_lower
thf(fact_6582_ccINF__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,B5: set @ C,F3: B > A,G3: C > A] :
( ( countable_countable @ B @ A6 )
=> ( ( countable_countable @ C @ B5 )
=> ( ! [M: C] :
( ( member @ C @ M @ B5 )
=> ? [X: B] :
( ( member @ B @ X @ A6 )
& ( ord_less_eq @ A @ ( F3 @ X ) @ ( G3 @ M ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G3 @ B5 ) ) ) ) ) ) ) ).
% ccINF_mono
thf(fact_6583_ccSup__inter__less__eq,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( countable_countable @ A @ A6 )
=> ( ( countable_countable @ A @ B5 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ) ) ).
% ccSup_inter_less_eq
thf(fact_6584_less__eq__ccInf__inter,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( countable_countable @ A @ A6 )
=> ( ( countable_countable @ A @ B5 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B5 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ) ) ) ) ).
% less_eq_ccInf_inter
thf(fact_6585_ccSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B5: set @ B,A6: set @ B,F3: B > A,G3: B > A] :
( ( countable_countable @ B @ B5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ B5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ B5 ) ) ) ) ) ) ) ).
% ccSUP_subset_mono
thf(fact_6586_ccINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,B5: set @ B,F3: B > A,G3: B > A] :
( ( countable_countable @ B @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ A6 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ B5 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B5 ) ) ) ) ) ) ) ).
% ccINF_superset_mono
thf(fact_6587_mono__ccSup,axiom,
! [B: $tType,A: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F3: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( countable_countable @ A @ A6 )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ A6 ) ) @ ( F3 @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ).
% mono_ccSup
thf(fact_6588_mono__ccSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F3: A > B,I5: set @ C,A6: C > A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( countable_countable @ C @ I5 )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F3 @ ( A6 @ X4 ) )
@ I5 ) )
@ ( F3 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A6 @ I5 ) ) ) ) ) ) ) ).
% mono_ccSUP
thf(fact_6589_mono__ccINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F3: A > B,I5: set @ C,A6: C > A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( countable_countable @ C @ I5 )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A6 @ I5 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F3 @ ( A6 @ X4 ) )
@ I5 ) ) ) ) ) ) ).
% mono_ccINF
thf(fact_6590_mono__ccInf,axiom,
! [B: $tType,A: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F3: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( countable_countable @ A @ A6 )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ A6 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ A6 ) ) ) ) ) ) ).
% mono_ccInf
thf(fact_6591_and__num_Opelims,axiom,
! [X3: num,Xa2: num,Y: option @ num] :
( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ X3 @ Xa2 ) )
=> ( ( ( X3 = one2 )
=> ( ( Xa2 = one2 )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa2
= ( bit0 @ N2 ) )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa2
= ( bit1 @ N2 ) )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ( ( Xa2 = one2 )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M ) @ one2 ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit0 @ N2 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit1 @ N2 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M ) @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ( ( Xa2 = one2 )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M ) @ one2 ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit0 @ N2 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ~ ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit1 @ N2 ) )
=> ( ( Y
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N10: num] : ( some @ num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
thf(fact_6592_xor__num_Opelims,axiom,
! [X3: num,Xa2: num,Y: option @ num] :
( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ X3 @ Xa2 ) )
=> ( ( ( X3 = one2 )
=> ( ( Xa2 = one2 )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa2
= ( bit0 @ N2 ) )
=> ( ( Y
= ( some @ num @ ( bit1 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa2
= ( bit1 @ N2 ) )
=> ( ( Y
= ( some @ num @ ( bit0 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ( ( Xa2 = one2 )
=> ( ( Y
= ( some @ num @ ( bit1 @ M ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M ) @ one2 ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit0 @ N2 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit1 @ N2 ) )
=> ( ( Y
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M ) @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ( ( Xa2 = one2 )
=> ( ( Y
= ( some @ num @ ( bit0 @ M ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M ) @ one2 ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit0 @ N2 ) )
=> ( ( Y
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ~ ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N2: num] :
( ( Xa2
= ( bit1 @ N2 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
thf(fact_6593_or__not__num__neg_Opelims,axiom,
! [X3: num,Xa2: num,Y: num] :
( ( ( bit_or_not_num_neg @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ X3 @ Xa2 ) )
=> ( ( ( X3 = one2 )
=> ( ( Xa2 = one2 )
=> ( ( Y = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [M: num] :
( ( Xa2
= ( bit0 @ M ) )
=> ( ( Y
= ( bit1 @ M ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ M ) ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [M: num] :
( ( Xa2
= ( bit1 @ M ) )
=> ( ( Y
= ( bit1 @ M ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ M ) ) ) ) ) )
=> ( ! [N2: num] :
( ( X3
= ( bit0 @ N2 ) )
=> ( ( Xa2 = one2 )
=> ( ( Y
= ( bit0 @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N2 ) @ one2 ) ) ) ) )
=> ( ! [N2: num] :
( ( X3
= ( bit0 @ N2 ) )
=> ! [M: num] :
( ( Xa2
= ( bit0 @ M ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N2 ) @ ( bit0 @ M ) ) ) ) ) )
=> ( ! [N2: num] :
( ( X3
= ( bit0 @ N2 ) )
=> ! [M: num] :
( ( Xa2
= ( bit1 @ M ) )
=> ( ( Y
= ( bit0 @ ( bit_or_not_num_neg @ N2 @ M ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N2 ) @ ( bit1 @ M ) ) ) ) ) )
=> ( ! [N2: num] :
( ( X3
= ( bit1 @ N2 ) )
=> ( ( Xa2 = one2 )
=> ( ( Y = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N2 ) @ one2 ) ) ) ) )
=> ( ! [N2: num] :
( ( X3
= ( bit1 @ N2 ) )
=> ! [M: num] :
( ( Xa2
= ( bit0 @ M ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N2 ) @ ( bit0 @ M ) ) ) ) ) )
=> ~ ! [N2: num] :
( ( X3
= ( bit1 @ N2 ) )
=> ! [M: num] :
( ( Xa2
= ( bit1 @ M ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N2 ) @ ( bit1 @ M ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
thf(fact_6594_minus__set__fold,axiom,
! [A: $tType,A6: set @ A,Xs2: list @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ ( set2 @ A @ Xs2 ) )
= ( fold @ A @ ( set @ A ) @ ( remove @ A ) @ Xs2 @ A6 ) ) ).
% minus_set_fold
thf(fact_6595_member__remove,axiom,
! [A: $tType,X3: A,Y: A,A6: set @ A] :
( ( member @ A @ X3 @ ( remove @ A @ Y @ A6 ) )
= ( ( member @ A @ X3 @ A6 )
& ( X3 != Y ) ) ) ).
% member_remove
thf(fact_6596_remove__code_I1_J,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( remove @ A @ X3 @ ( set2 @ A @ Xs2 ) )
= ( set2 @ A @ ( removeAll @ A @ X3 @ Xs2 ) ) ) ).
% remove_code(1)
thf(fact_6597_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X4: A,A7: set @ A] : ( minus_minus @ ( set @ A ) @ A7 @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_6598_range__from__nat__into,axiom,
! [A: $tType,A6: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( countable_countable @ A @ A6 )
=> ( ( image2 @ nat @ A @ ( counta4804993851260445106t_into @ A @ A6 ) @ ( top_top @ ( set @ nat ) ) )
= A6 ) ) ) ).
% range_from_nat_into
thf(fact_6599_Field__insert,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 ) )
= ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) ) ) ).
% Field_insert
thf(fact_6600_Field__empty,axiom,
! [A: $tType] :
( ( field2 @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Field_empty
thf(fact_6601_Field__Un,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S ) )
= ( sup_sup @ ( set @ A ) @ ( field2 @ A @ R2 ) @ ( field2 @ A @ S ) ) ) ).
% Field_Un
thf(fact_6602_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_6603_from__nat__into__inject,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( countable_countable @ A @ A6 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( countable_countable @ A @ B5 )
=> ( ( ( counta4804993851260445106t_into @ A @ A6 )
= ( counta4804993851260445106t_into @ A @ B5 ) )
= ( A6 = B5 ) ) ) ) ) ) ).
% from_nat_into_inject
thf(fact_6604_from__nat__into,axiom,
! [A: $tType,A6: set @ A,N: nat] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( counta4804993851260445106t_into @ A @ A6 @ N ) @ A6 ) ) ).
% from_nat_into
thf(fact_6605_mono__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
=> ( ord_less_eq @ ( set @ A ) @ ( field2 @ A @ R2 ) @ ( field2 @ A @ S ) ) ) ).
% mono_Field
thf(fact_6606_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_6607_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_6608_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_6609_inj__on__from__nat__into,axiom,
! [A: $tType] :
( inj_on @ ( set @ A ) @ ( nat > A ) @ ( counta4804993851260445106t_into @ A )
@ ( collect @ ( set @ A )
@ ^ [A7: set @ A] :
( ( A7
!= ( bot_bot @ ( set @ A ) ) )
& ( countable_countable @ A @ A7 ) ) ) ) ).
% inj_on_from_nat_into
thf(fact_6610_range__from__nat__into__subset,axiom,
! [A: $tType,A6: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ ( counta4804993851260445106t_into @ A @ A6 ) @ ( top_top @ ( set @ nat ) ) ) @ A6 ) ) ).
% range_from_nat_into_subset
thf(fact_6611_subset__range__from__nat__into,axiom,
! [A: $tType,A6: set @ A] :
( ( countable_countable @ A @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ ( image2 @ nat @ A @ ( counta4804993851260445106t_into @ A @ A6 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% subset_range_from_nat_into
thf(fact_6612_Total__subset__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) )
=> ( ( R2
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
| ? [A5: A] :
( R2
= ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ).
% Total_subset_Id
thf(fact_6613_Field__natLeq__on,axiom,
! [N: nat] :
( ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y3: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X4 @ Y3 ) ) ) ) )
= ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N ) ) ) ).
% Field_natLeq_on
thf(fact_6614_UnderS__def,axiom,
! [A: $tType] :
( ( order_UnderS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A7: set @ A] :
( collect @ A
@ ^ [B8: A] :
( ( member @ A @ B8 @ ( field2 @ A @ R5 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ( B8 != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% UnderS_def
thf(fact_6615_Under__def,axiom,
! [A: $tType] :
( ( order_Under @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A7: set @ A] :
( collect @ A
@ ^ [B8: A] :
( ( member @ A @ B8 @ ( field2 @ A @ R5 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ X4 ) @ R5 ) ) ) ) ) ) ).
% Under_def
thf(fact_6616_Above__def,axiom,
! [A: $tType] :
( ( order_Above @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A7: set @ A] :
( collect @ A
@ ^ [B8: A] :
( ( member @ A @ B8 @ ( field2 @ A @ R5 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ B8 ) @ R5 ) ) ) ) ) ) ).
% Above_def
thf(fact_6617_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ A ),As3: A > B] :
! [I4: A,J3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I4 @ J3 ) @ R5 )
=> ( ord_less_eq @ B @ ( As3 @ I4 ) @ ( As3 @ J3 ) ) ) ) ) ) ).
% relChain_def
thf(fact_6618_cofinal__def,axiom,
! [A: $tType] :
( ( bNF_Ca7293521722713021262ofinal @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R5 ) )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A7 )
& ( X4 != Y3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R5 ) ) ) ) ) ).
% cofinal_def
thf(fact_6619_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
@ ^ [B15: A,B24: A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A12 @ ( insert2 @ A @ A23 @ ( insert2 @ A @ B15 @ ( insert2 @ A @ B24 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R5 ) )
& ( ( ( A12 = B15 )
& ( A23 = B24 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A12 @ A23 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B15 @ B24 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B15 @ B24 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B15 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B15 @ B24 ) )
& ( A12 = B15 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A23 @ B24 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_6620_linear__order__on__singleton,axiom,
! [A: $tType,X3: A] : ( order_679001287576687338der_on @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% linear_order_on_singleton
thf(fact_6621_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
! [A: $tType,B: $tType,S3: set @ A,F3: A > B > B,A6: set @ A,Z2: B,Y: B,A3: A] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ S3 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A6 @ Y )
=> ( ( member @ A @ A3 @ A6 )
=> ? [Y16: B] :
( ( Y
= ( F3 @ A3 @ Y16 ) )
& ( finite_fold_graph @ A @ B @ F3 @ Z2 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ Y16 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE_aux
thf(fact_6622_fold__graph_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( finite_fold_graph @ A @ B )
= ( ^ [F4: A > B > B,Z4: B,A12: set @ A,A23: B] :
( ( ( A12
= ( bot_bot @ ( set @ A ) ) )
& ( A23 = Z4 ) )
| ? [X4: A,A7: set @ A,Y3: B] :
( ( A12
= ( insert2 @ A @ X4 @ A7 ) )
& ( A23
= ( F4 @ X4 @ Y3 ) )
& ~ ( member @ A @ X4 @ A7 )
& ( finite_fold_graph @ A @ B @ F4 @ Z4 @ A7 @ Y3 ) ) ) ) ) ).
% fold_graph.simps
thf(fact_6623_fold__graph_Ocases,axiom,
! [A: $tType,B: $tType,F3: A > B > B,Z2: B,A1: set @ A,A22: B] :
( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A1 @ A22 )
=> ( ( ( A1
= ( bot_bot @ ( set @ A ) ) )
=> ( A22 != Z2 ) )
=> ~ ! [X5: A,A10: set @ A] :
( ( A1
= ( insert2 @ A @ X5 @ A10 ) )
=> ! [Y4: B] :
( ( A22
= ( F3 @ X5 @ Y4 ) )
=> ( ~ ( member @ A @ X5 @ A10 )
=> ~ ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A10 @ Y4 ) ) ) ) ) ) ).
% fold_graph.cases
thf(fact_6624_fold__graph_OinsertI,axiom,
! [A: $tType,B: $tType,X3: A,A6: set @ A,F3: A > B > B,Z2: B,Y: B] :
( ~ ( member @ A @ X3 @ A6 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A6 @ Y )
=> ( finite_fold_graph @ A @ B @ F3 @ Z2 @ ( insert2 @ A @ X3 @ A6 ) @ ( F3 @ X3 @ Y ) ) ) ) ).
% fold_graph.insertI
thf(fact_6625_empty__fold__graphE,axiom,
! [A: $tType,B: $tType,F3: A > B > B,Z2: B,X3: B] :
( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ ( bot_bot @ ( set @ A ) ) @ X3 )
=> ( X3 = Z2 ) ) ).
% empty_fold_graphE
thf(fact_6626_fold__graph_OemptyI,axiom,
! [A: $tType,B: $tType,F3: A > B > B,Z2: B] : ( finite_fold_graph @ A @ B @ F3 @ Z2 @ ( bot_bot @ ( set @ A ) ) @ Z2 ) ).
% fold_graph.emptyI
thf(fact_6627_comp__fun__commute__on_Ofold__graph__determ,axiom,
! [A: $tType,B: $tType,S3: set @ A,F3: A > B > B,A6: set @ A,Z2: B,X3: B,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ S3 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A6 @ X3 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A6 @ Y )
=> ( Y = X3 ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_determ
thf(fact_6628_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_6629_comp__fun__commute__on_Ofold__graph__insertE,axiom,
! [A: $tType,B: $tType,S3: set @ A,F3: A > B > B,X3: A,A6: set @ A,Z2: B,V2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X3 @ A6 ) @ S3 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ ( insert2 @ A @ X3 @ A6 ) @ V2 )
=> ( ~ ( member @ A @ X3 @ A6 )
=> ~ ! [Y4: B] :
( ( V2
= ( F3 @ X3 @ Y4 ) )
=> ~ ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A6 @ Y4 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE
thf(fact_6630_comp__fun__commute__on_Ofold__equality,axiom,
! [A: $tType,B: $tType,S3: set @ A,F3: A > B > B,A6: set @ A,Z2: B,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ S3 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A6 @ Y )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ A6 )
= Y ) ) ) ) ).
% comp_fun_commute_on.fold_equality
thf(fact_6631_comp__fun__commute__on_Ofold__graph__fold,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,A6: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ S3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A6 @ ( finite_fold @ A @ B @ F3 @ Z2 @ A6 ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_fold
thf(fact_6632_Linear__order__in__diff__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_6633_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) )
= ( ! [A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R2 ) )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_6634_increasing__Bseq__subseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G3: nat > nat] :
( ! [X5: nat,Y4: nat] :
( ( ord_less_eq @ nat @ X5 @ Y4 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X5 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ Y4 ) ) ) )
=> ( ( order_strict_mono @ nat @ nat @ G3 )
=> ( ( bfun @ nat @ A
@ ^ [X4: nat] : ( F3 @ ( G3 @ X4 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ) ) ).
% increasing_Bseq_subseq_iff
thf(fact_6635_wf__insert,axiom,
! [A: $tType,Y: A,X3: A,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X3 ) @ R2 ) )
= ( ( wf @ A @ R2 )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% wf_insert
thf(fact_6636_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_6637_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 )
=> ~ ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z3 ) @ R )
=> ~ ( member @ A @ Y6 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_6638_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ( F3 @ X3 )
= ( F3 @ Y ) )
= ( X3 = Y ) ) ) ) ).
% strict_mono_eq
thf(fact_6639_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ) ) ).
% strict_monoD
thf(fact_6640_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X5: A,Y4: A] :
( ( ord_less @ A @ X5 @ Y4 )
=> ( ord_less @ B @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) )
=> ( order_strict_mono @ A @ B @ F3 ) ) ) ).
% strict_monoI
thf(fact_6641_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F4: A > B] :
! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less @ B @ ( F4 @ X4 ) @ ( F4 @ Y3 ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_6642_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) )
= ( ord_less @ A @ X3 @ Y ) ) ) ) ).
% strict_mono_less
thf(fact_6643_wf__induct__rule,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( wf @ A @ R2 )
=> ( ! [X5: A] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X5 ) @ R2 )
=> ( P @ Y6 ) )
=> ( P @ X5 ) )
=> ( P @ A3 ) ) ) ).
% wf_induct_rule
thf(fact_6644_wf__eq__minimal,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [Q6: set @ A] :
( ? [X4: A] : ( member @ A @ X4 @ Q6 )
=> ? [X4: A] :
( ( member @ A @ X4 @ Q6 )
& ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R5 )
=> ~ ( member @ A @ Y3 @ Q6 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_6645_wf__not__refl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( wf @ A @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R2 ) ) ).
% wf_not_refl
thf(fact_6646_wf__not__sym,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,X3: A] :
( ( wf @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X3 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A3 ) @ R2 ) ) ) ).
% wf_not_sym
thf(fact_6647_wf__irrefl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( wf @ A @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R2 ) ) ).
% wf_irrefl
thf(fact_6648_wf__induct,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( wf @ A @ R2 )
=> ( ! [X5: A] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X5 ) @ R2 )
=> ( P @ Y6 ) )
=> ( P @ X5 ) )
=> ( P @ A3 ) ) ) ).
% wf_induct
thf(fact_6649_wf__asym,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,X3: A] :
( ( wf @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X3 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A3 ) @ R2 ) ) ) ).
% wf_asym
thf(fact_6650_wfUNIVI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [P7: A > $o,X5: A] :
( ! [Xa: A] :
( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa ) @ R2 )
=> ( P7 @ Y4 ) )
=> ( P7 @ Xa ) )
=> ( P7 @ X5 ) )
=> ( wf @ A @ R2 ) ) ).
% wfUNIVI
thf(fact_6651_wfI__min,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X5: A,Q7: set @ A] :
( ( member @ A @ X5 @ Q7 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Q7 )
& ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa ) @ R )
=> ~ ( member @ A @ Y4 @ Q7 ) ) ) )
=> ( wf @ A @ R ) ) ).
% wfI_min
thf(fact_6652_wfE__min,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X3: A,Q: set @ A] :
( ( wf @ A @ R )
=> ( ( member @ A @ X3 @ Q )
=> ~ ! [Z3: A] :
( ( member @ A @ Z3 @ Q )
=> ~ ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z3 ) @ R )
=> ~ ( member @ A @ Y6 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_6653_wf__def,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [P4: A > $o] :
( ! [X4: A] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R5 )
=> ( P4 @ Y3 ) )
=> ( P4 @ X4 ) )
=> ! [X8: A] : ( P4 @ X8 ) ) ) ) ).
% wf_def
thf(fact_6654_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
~ ? [F4: nat > A] :
! [I4: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F4 @ ( suc @ I4 ) ) @ ( F4 @ I4 ) ) @ R5 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_6655_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),F3: nat > A] :
( ( wf @ A @ R2 )
=> ~ ! [K: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F3 @ ( suc @ K ) ) @ ( F3 @ K ) ) @ R2 ) ) ).
% wf_no_infinite_down_chainE
thf(fact_6656_strict__mono__add,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A] :
( order_strict_mono @ A @ A
@ ^ [N3: A] : ( plus_plus @ A @ N3 @ K2 ) ) ) ).
% strict_mono_add
thf(fact_6657_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_6658_strict__mono__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [R2: A > B,M2: A,N: A] :
( ( order_strict_mono @ A @ B @ R2 )
=> ( ( ord_less_eq @ A @ M2 @ N )
=> ( ord_less_eq @ B @ ( R2 @ M2 ) @ ( R2 @ N ) ) ) ) ) ).
% strict_mono_leD
thf(fact_6659_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) )
= ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ).
% strict_mono_less_eq
thf(fact_6660_strict__mono__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_strict_mono @ nat @ A )
= ( ^ [F4: nat > A] :
! [N3: nat] : ( ord_less @ A @ ( F4 @ N3 ) @ ( F4 @ ( suc @ N3 ) ) ) ) ) ) ).
% strict_mono_Suc_iff
thf(fact_6661_wf__bounded__measure,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > nat,F3: A > nat] :
( ! [A5: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R2 )
=> ( ( ord_less_eq @ nat @ ( Ub @ B4 ) @ ( Ub @ A5 ) )
& ( ord_less_eq @ nat @ ( F3 @ B4 ) @ ( Ub @ A5 ) )
& ( ord_less @ nat @ ( F3 @ A5 ) @ ( F3 @ B4 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wf_bounded_measure
thf(fact_6662_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P: B > $o,K2: B,M2: B > A] :
( ( wf @ A @ R2 )
=> ( ! [X5: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ ( transitive_trancl @ A @ R2 ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X5 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) )
=> ( ( P @ K2 )
=> ? [X5: B] :
( ( P @ X5 )
& ! [Y6: B] :
( ( P @ Y6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M2 @ X5 ) @ ( M2 @ Y6 ) ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_6663_wf__eq__minimal2,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [A7: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R5 ) )
& ( A7
!= ( bot_bot @ ( set @ A ) ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A7 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R5 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_6664_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F3: A > ( set @ B )] :
( ! [A5: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R2 )
=> ( ( finite_finite2 @ B @ ( Ub @ A5 ) )
& ( ord_less_eq @ ( set @ B ) @ ( Ub @ B4 ) @ ( Ub @ A5 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F3 @ B4 ) @ ( Ub @ A5 ) )
& ( ord_less @ ( set @ B ) @ ( F3 @ A5 ) @ ( F3 @ B4 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wf_bounded_set
thf(fact_6665_finite__subset__wf,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( wf @ ( set @ A )
@ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X8: set @ A,Y10: set @ A] :
( ( ord_less @ ( set @ A ) @ X8 @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ Y10 @ A6 ) ) ) ) ) ) ).
% finite_subset_wf
thf(fact_6666_reduction__pairI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S3 ) @ R )
=> ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ S3 ) ) ) ) ).
% reduction_pairI
thf(fact_6667_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P: ( A > B ) > A > B > $o] :
( ( wf @ A @ R )
=> ( ! [F2: A > B,G2: A > B,X5: A,R3: B] :
( ! [Z5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z5 @ X5 ) @ R )
=> ( ( F2 @ Z5 )
= ( G2 @ Z5 ) ) )
=> ( ( P @ F2 @ X5 @ R3 )
= ( P @ G2 @ X5 @ R3 ) ) )
=> ( ! [X5: A,F2: A > B] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X5 ) @ R )
=> ( P @ F2 @ Y6 @ ( F2 @ Y6 ) ) )
=> ? [X_1: B] : ( P @ F2 @ X5 @ X_1 ) )
=> ? [F2: A > B] :
! [X: A] : ( P @ F2 @ X @ ( F2 @ X ) ) ) ) ) ).
% dependent_wf_choice
thf(fact_6668_chains__extend,axiom,
! [A: $tType,C3: set @ ( set @ A ),S3: set @ ( set @ A ),Z2: set @ A] :
( ( member @ ( set @ ( set @ A ) ) @ C3 @ ( chains2 @ A @ S3 ) )
=> ( ( member @ ( set @ A ) @ Z2 @ S3 )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ X5 @ Z2 ) )
=> ( member @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert2 @ ( set @ A ) @ Z2 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C3 ) @ ( chains2 @ A @ S3 ) ) ) ) ) ).
% chains_extend
thf(fact_6669_finite__def,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( complete_lattice_lfp @ ( ( set @ A ) > $o )
@ ^ [P5: ( set @ A ) > $o,X4: set @ A] :
( ( X4
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,A8: A] :
( ( X4
= ( insert2 @ A @ A8 @ A7 ) )
& ( P5 @ A7 ) ) ) ) ) ).
% finite_def
thf(fact_6670_Zorn__Lemma2,axiom,
! [A: $tType,A6: set @ ( set @ A )] :
( ! [X5: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X5 @ ( chains2 @ A @ A6 ) )
=> ? [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A6 )
& ! [Xb2: set @ A] :
( ( member @ ( set @ A ) @ Xb2 @ X5 )
=> ( ord_less_eq @ ( set @ A ) @ Xb2 @ Xa ) ) ) )
=> ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A6 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ Xa )
=> ( Xa = X5 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_6671_chainsD,axiom,
! [A: $tType,C3: set @ ( set @ A ),S3: set @ ( set @ A ),X3: set @ A,Y: set @ A] :
( ( member @ ( set @ ( set @ A ) ) @ C3 @ ( chains2 @ A @ S3 ) )
=> ( ( member @ ( set @ A ) @ X3 @ C3 )
=> ( ( member @ ( set @ A ) @ Y @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Y )
| ( ord_less_eq @ ( set @ A ) @ Y @ X3 ) ) ) ) ) ).
% chainsD
thf(fact_6672_lfp__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F6: A > A,X3: A] :
( ( order_mono @ A @ A @ F6 )
=> ( ( ( F6 @ X3 )
= X3 )
=> ( ! [Z3: A] :
( ( ( F6 @ Z3 )
= Z3 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ( complete_lattice_lfp @ A @ F6 )
= X3 ) ) ) ) ) ).
% lfp_eqI
thf(fact_6673_lfp__lfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A > A] :
( ! [X5: A,Y4: A,W2: A,Z3: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ( ord_less_eq @ A @ W2 @ Z3 )
=> ( ord_less_eq @ A @ ( F3 @ X5 @ W2 ) @ ( F3 @ Y4 @ Z3 ) ) ) )
=> ( ( complete_lattice_lfp @ A
@ ^ [X4: A] : ( complete_lattice_lfp @ A @ ( F3 @ X4 ) ) )
= ( complete_lattice_lfp @ A
@ ^ [X4: A] : ( F3 @ X4 @ X4 ) ) ) ) ) ).
% lfp_lfp
thf(fact_6674_lfp__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,A6: A] :
( ! [U4: A] :
( ( ord_less_eq @ A @ ( F3 @ U4 ) @ U4 )
=> ( ord_less_eq @ A @ A6 @ U4 ) )
=> ( ord_less_eq @ A @ A6 @ ( complete_lattice_lfp @ A @ F3 ) ) ) ) ).
% lfp_greatest
thf(fact_6675_lfp__lowerbound,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,A6: A] :
( ( ord_less_eq @ A @ ( F3 @ A6 ) @ A6 )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F3 ) @ A6 ) ) ) ).
% lfp_lowerbound
thf(fact_6676_lfp__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,G3: A > A] :
( ! [Z10: A] : ( ord_less_eq @ A @ ( F3 @ Z10 ) @ ( G3 @ Z10 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F3 ) @ ( complete_lattice_lfp @ A @ G3 ) ) ) ) ).
% lfp_mono
thf(fact_6677_lfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_lfp @ A )
= ( ^ [F4: A > A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [U2: A] : ( ord_less_eq @ A @ ( F4 @ U2 ) @ U2 ) ) ) ) ) ) ).
% lfp_def
thf(fact_6678_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_6679_def__lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: A,F3: A > A,P: A] :
( ( A6
= ( complete_lattice_lfp @ A @ F3 ) )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ ( inf_inf @ A @ A6 @ P ) ) @ P )
=> ( ord_less_eq @ A @ A6 @ P ) ) ) ) ) ).
% def_lfp_induct
thf(fact_6680_lfp__ordinal__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,P: A > $o] :
( ( order_mono @ A @ A @ F3 )
=> ( ! [S4: A] :
( ( P @ S4 )
=> ( ( ord_less_eq @ A @ S4 @ ( complete_lattice_lfp @ A @ F3 ) )
=> ( P @ ( F3 @ S4 ) ) ) )
=> ( ! [M8: set @ A] :
( ! [X: A] :
( ( member @ A @ X @ M8 )
=> ( P @ X ) )
=> ( P @ ( complete_Sup_Sup @ A @ M8 ) ) )
=> ( P @ ( complete_lattice_lfp @ A @ F3 ) ) ) ) ) ) ).
% lfp_ordinal_induct
thf(fact_6681_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_6682_Zorn__Lemma,axiom,
! [A: $tType,A6: set @ ( set @ A )] :
( ! [X5: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X5 @ ( chains2 @ A @ A6 ) )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ X5 ) @ A6 ) )
=> ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A6 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ Xa )
=> ( Xa = X5 ) ) ) ) ) ).
% Zorn_Lemma
thf(fact_6683_lfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,K2: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K2 ) @ F3 @ ( bot_bot @ A ) )
= ( compow @ ( A > A ) @ K2 @ F3 @ ( bot_bot @ A ) ) )
=> ( ( complete_lattice_lfp @ A @ F3 )
= ( compow @ ( A > A ) @ K2 @ F3 @ ( bot_bot @ A ) ) ) ) ) ) ).
% lfp_Kleene_iter
thf(fact_6684_iteratesp__def,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F4: A > A] :
( complete_lattice_lfp @ ( A > $o )
@ ^ [P5: A > $o,X4: A] :
( ? [Y3: A] :
( ( X4
= ( F4 @ Y3 ) )
& ( P5 @ Y3 ) )
| ? [M9: set @ A] :
( ( X4
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ M9 )
=> ( P5 @ Y3 ) ) ) ) ) ) ) ) ).
% iteratesp_def
thf(fact_6685_lfp__transfer__bounded,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [P: A > $o,F3: A > A,Alpha: A > B,G3: B > B] :
( ( P @ ( bot_bot @ A ) )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( P @ ( F3 @ X5 ) ) )
=> ( ! [M8: nat > A] :
( ! [I2: nat] : ( P @ ( M8 @ I2 ) )
=> ( P @ ( complete_Sup_Sup @ A @ ( image2 @ nat @ A @ M8 @ ( top_top @ ( set @ nat ) ) ) ) ) )
=> ( ! [M8: nat > A] :
( ( order_mono @ nat @ A @ M8 )
=> ( ! [I2: nat] : ( P @ ( M8 @ I2 ) )
=> ( ( Alpha @ ( complete_Sup_Sup @ A @ ( image2 @ nat @ A @ M8 @ ( top_top @ ( set @ nat ) ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ nat @ B
@ ^ [I4: nat] : ( Alpha @ ( M8 @ I4 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) )
=> ( ( order_sup_continuous @ A @ A @ F3 )
=> ( ( order_sup_continuous @ B @ B @ G3 )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( ( ord_less_eq @ A @ X5 @ ( complete_lattice_lfp @ A @ F3 ) )
=> ( ( Alpha @ ( F3 @ X5 ) )
= ( G3 @ ( Alpha @ X5 ) ) ) ) )
=> ( ! [X5: B] : ( ord_less_eq @ B @ ( Alpha @ ( bot_bot @ A ) ) @ ( G3 @ X5 ) )
=> ( ( Alpha @ ( complete_lattice_lfp @ A @ F3 ) )
= ( complete_lattice_lfp @ B @ G3 ) ) ) ) ) ) ) ) ) ) ) ).
% lfp_transfer_bounded
thf(fact_6686_lfp__induct2,axiom,
! [A: $tType,B: $tType,A3: A,B2: 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 @ B2 ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F3 ) )
=> ( ( order_mono @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ ( product_prod @ A @ B ) ) @ F3 )
=> ( ! [A5: A,B4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ ( F3 @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) ) ) )
=> ( P @ A5 @ B4 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% lfp_induct2
thf(fact_6687_lfp__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [Alpha: A > B,F3: A > A,G3: B > B] :
( ( order_sup_continuous @ A @ B @ Alpha )
=> ( ( order_sup_continuous @ A @ A @ F3 )
=> ( ( order_sup_continuous @ B @ B @ G3 )
=> ( ! [X5: B] : ( ord_less_eq @ B @ ( Alpha @ ( bot_bot @ A ) ) @ ( G3 @ X5 ) )
=> ( ! [X5: A] :
( ( ord_less_eq @ A @ X5 @ ( complete_lattice_lfp @ A @ F3 ) )
=> ( ( Alpha @ ( F3 @ X5 ) )
= ( G3 @ ( Alpha @ X5 ) ) ) )
=> ( ( Alpha @ ( complete_lattice_lfp @ A @ F3 ) )
= ( complete_lattice_lfp @ B @ G3 ) ) ) ) ) ) ) ) ).
% lfp_transfer
thf(fact_6688_cclfp__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( counta3822494911875563373attice @ B )
& ( counta3822494911875563373attice @ A ) )
=> ! [Alpha: A > B,F3: A > A,G3: B > B] :
( ( order_sup_continuous @ A @ B @ Alpha )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ( ( Alpha @ ( bot_bot @ A ) )
= ( bot_bot @ B ) )
=> ( ! [X5: A] :
( ( Alpha @ ( F3 @ X5 ) )
= ( G3 @ ( Alpha @ X5 ) ) )
=> ( ( Alpha @ ( order_532582986084564980_cclfp @ A @ F3 ) )
= ( order_532582986084564980_cclfp @ B @ G3 ) ) ) ) ) ) ) ).
% cclfp_transfer
thf(fact_6689_iteratesp_OSup,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [M7: set @ A,F3: A > A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M7 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ M7 )
=> ( comple7512665784863727008ratesp @ A @ F3 @ X5 ) )
=> ( comple7512665784863727008ratesp @ A @ F3 @ ( complete_Sup_Sup @ A @ M7 ) ) ) ) ) ).
% iteratesp.Sup
thf(fact_6690_iteratesp_Ocases,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F3: A > A,A3: A] :
( ( comple7512665784863727008ratesp @ A @ F3 @ A3 )
=> ( ! [X5: A] :
( ( A3
= ( F3 @ X5 ) )
=> ~ ( comple7512665784863727008ratesp @ A @ F3 @ X5 ) )
=> ~ ! [M8: set @ A] :
( ( A3
= ( complete_Sup_Sup @ A @ M8 ) )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M8 )
=> ~ ! [X: A] :
( ( member @ A @ X @ M8 )
=> ( comple7512665784863727008ratesp @ A @ F3 @ X ) ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_6691_iteratesp_Osimps,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F4: A > A,A8: A] :
( ? [X4: A] :
( ( A8
= ( F4 @ X4 ) )
& ( comple7512665784863727008ratesp @ A @ F4 @ X4 ) )
| ? [M9: set @ A] :
( ( A8
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [X4: A] :
( ( member @ A @ X4 @ M9 )
=> ( comple7512665784863727008ratesp @ A @ F4 @ X4 ) ) ) ) ) ) ) ).
% iteratesp.simps
thf(fact_6692_sup__continuous__lfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F6: A > A] :
( ( order_sup_continuous @ A @ A @ F6 )
=> ( ( complete_lattice_lfp @ A @ F6 )
= ( complete_Sup_Sup @ A
@ ( image2 @ nat @ A
@ ^ [I4: nat] : ( compow @ ( A > A ) @ I4 @ F6 @ ( bot_bot @ A ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% sup_continuous_lfp
thf(fact_6693_finite__refines__card__le,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A6 @ R ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S3 )
=> ( ( equiv_equiv @ A @ A6 @ R )
=> ( ( equiv_equiv @ A @ A6 @ S3 )
=> ( ord_less_eq @ nat @ ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A6 @ S3 ) ) @ ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A6 @ R ) ) ) ) ) ) ) ).
% finite_refines_card_le
thf(fact_6694_butlast__take,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( butlast @ A @ ( take @ A @ N @ Xs2 ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs2 ) ) ) ).
% butlast_take
thf(fact_6695_length__butlast,axiom,
! [A: $tType,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs2 ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) ) ).
% length_butlast
thf(fact_6696_quotient__eqI,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),X6: set @ A,Y8: set @ A,X3: A,Y: A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y8 @ ( equiv_quotient @ A @ A6 @ R2 ) )
=> ( ( member @ A @ X3 @ X6 )
=> ( ( member @ A @ Y @ Y8 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( X6 = Y8 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_6697_quotient__eq__iff,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),X6: set @ A,Y8: set @ A,X3: A,Y: A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y8 @ ( equiv_quotient @ A @ A6 @ R2 ) )
=> ( ( member @ A @ X3 @ X6 )
=> ( ( member @ A @ Y @ Y8 )
=> ( ( X6 = Y8 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_6698_in__quotient__imp__closed,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),X6: set @ A,X3: A,Y: A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R2 ) )
=> ( ( member @ A @ X3 @ X6 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( member @ A @ Y @ X6 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_6699_in__set__butlast__appendI,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Ys: list @ A] :
( ( ( member @ A @ X3 @ ( set2 @ A @ ( butlast @ A @ Xs2 ) ) )
| ( member @ A @ X3 @ ( set2 @ A @ ( butlast @ A @ Ys ) ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ ( butlast @ A @ ( append @ A @ Xs2 @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_6700_in__quotient__imp__non__empty,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),X6: set @ A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R2 ) )
=> ( X6
!= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_6701_in__quotient__imp__subset,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),X6: set @ A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R2 ) )
=> ( ord_less_eq @ ( set @ A ) @ X6 @ A6 ) ) ) ).
% in_quotient_imp_subset
thf(fact_6702_in__set__butlastD,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( butlast @ A @ Xs2 ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) ) ) ).
% in_set_butlastD
thf(fact_6703_quotient__disj,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),X6: set @ A,Y8: set @ A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y8 @ ( equiv_quotient @ A @ A6 @ R2 ) )
=> ( ( X6 = Y8 )
| ( ( inf_inf @ ( set @ A ) @ X6 @ Y8 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% quotient_disj
thf(fact_6704_nth__butlast,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs2 ) ) )
=> ( ( nth @ A @ ( butlast @ A @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ N ) ) ) ).
% nth_butlast
thf(fact_6705_sorted__butlast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( butlast @ A @ Xs2 ) ) ) ) ) ).
% sorted_butlast
thf(fact_6706_take__butlast,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( take @ A @ N @ ( butlast @ A @ Xs2 ) )
= ( take @ A @ N @ Xs2 ) ) ) ).
% take_butlast
thf(fact_6707_butlast__power,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( compow @ ( ( list @ A ) > ( list @ A ) ) @ N @ ( butlast @ A ) @ Xs2 )
= ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) @ Xs2 ) ) ).
% butlast_power
thf(fact_6708_eq__equiv__class__iff2,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( member @ A @ Y @ A6 )
=> ( ( ( equiv_quotient @ A @ ( insert2 @ A @ X3 @ ( 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 @ X3 @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_6709_butlast__conv__take,axiom,
! [A: $tType] :
( ( butlast @ A )
= ( ^ [Xs: list @ A] : ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% butlast_conv_take
thf(fact_6710_butlast__list__update,axiom,
! [A: $tType,K2: nat,Xs2: list @ A,X3: A] :
( ( ( K2
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs2 @ K2 @ X3 ) )
= ( butlast @ A @ Xs2 ) ) )
& ( ( K2
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs2 @ K2 @ X3 ) )
= ( list_update @ A @ ( butlast @ A @ Xs2 ) @ K2 @ X3 ) ) ) ) ).
% butlast_list_update
thf(fact_6711_in__quotient__imp__in__rel,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),X6: set @ A,X3: A,Y: A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ X6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_6712_UN__equiv__class__inject,axiom,
! [B: $tType,A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > ( set @ B ),X6: set @ A,Y8: set @ A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R2 @ F3 )
=> ( ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ X6 ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ Y8 ) ) )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y8 @ ( equiv_quotient @ A @ A6 @ R2 ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ A6 )
=> ( ( member @ A @ Y4 @ A6 )
=> ( ( ( F3 @ X5 )
= ( F3 @ Y4 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ R2 ) ) ) )
=> ( X6 = Y8 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
thf(fact_6713_proj__iff,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ A6 )
=> ( ( ( equiv_proj @ A @ A @ R2 @ X3 )
= ( equiv_proj @ A @ A @ R2 @ Y ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) ) ) ) ).
% proj_iff
thf(fact_6714_congruentI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F3: A > B] :
( ! [Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 )
=> ( ( F3 @ Y4 )
= ( F3 @ Z3 ) ) )
=> ( equiv_congruent @ A @ B @ R2 @ F3 ) ) ).
% congruentI
thf(fact_6715_congruentD,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F3: A > B,Y: A,Z2: A] :
( ( equiv_congruent @ A @ B @ R2 @ F3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R2 )
=> ( ( F3 @ Y )
= ( F3 @ Z2 ) ) ) ) ).
% congruentD
thf(fact_6716_UN__equiv__class,axiom,
! [B: $tType,A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > ( set @ B ),A3: A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R2 @ F3 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ( 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_6717_power__int__numeral__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [M2: num,N: num] :
( ( power_int @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( inverse_inverse @ A @ ( numeral_numeral @ A @ ( pow @ M2 @ N ) ) ) ) ) ).
% power_int_numeral_neg_numeral
thf(fact_6718_ImageI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R2: set @ ( product_prod @ A @ B ),A6: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R2 )
=> ( ( member @ A @ A3 @ A6 )
=> ( member @ B @ B2 @ ( image @ A @ B @ R2 @ A6 ) ) ) ) ).
% ImageI
thf(fact_6719_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_6720_Image__Id,axiom,
! [A: $tType,A6: set @ A] :
( ( image @ A @ A @ ( id2 @ A ) @ A6 )
= A6 ) ).
% Image_Id
thf(fact_6721_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,W: num,M2: int] :
( ( power_int @ A @ ( times_times @ A @ X3 @ ( numeral_numeral @ A @ W ) ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ X3 @ M2 ) @ ( power_int @ A @ ( numeral_numeral @ A @ W ) @ M2 ) ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_6722_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [W: num,Y: A,M2: int] :
( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Y ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W ) @ M2 ) @ ( power_int @ A @ Y @ M2 ) ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_6723_power__int__of__nat,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ! [X3: A,N: nat] :
( ( power_int @ A @ X3 @ ( semiring_1_of_nat @ int @ N ) )
= ( power_power @ A @ X3 @ N ) ) ) ).
% power_int_of_nat
thf(fact_6724_Image__empty1,axiom,
! [B: $tType,A: $tType,X6: set @ B] :
( ( image @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) @ X6 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Image_empty1
thf(fact_6725_power__int__mult__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M2: num,N: num] :
( ( power_int @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( power_int @ A @ X3 @ ( numeral_numeral @ int @ ( times_times @ num @ M2 @ N ) ) ) ) ) ).
% power_int_mult_numeral
thf(fact_6726_Image__Id__on,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( image @ A @ A @ ( id_on @ A @ A6 ) @ B5 )
= ( inf_inf @ ( set @ A ) @ A6 @ B5 ) ) ).
% Image_Id_on
thf(fact_6727_Image__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: B > A > $o,A6: set @ B] :
( ( image @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) @ A6 )
= ( collect @ A
@ ^ [Y3: A] :
? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( P @ X4 @ Y3 ) ) ) ) ).
% Image_Collect_case_prod
thf(fact_6728_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B2: A,R2: set @ ( product_prod @ B @ A ),A3: B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_6729_power__int__numeral,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ! [X3: A,N: num] :
( ( power_int @ A @ X3 @ ( numeral_numeral @ int @ N ) )
= ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% power_int_numeral
thf(fact_6730_of__real__eq__numeral__power__int__cancel__iff,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [Y: real,X3: num,N: int] :
( ( ( real_Vector_of_real @ A @ Y )
= ( power_int @ A @ ( numeral_numeral @ A @ X3 ) @ N ) )
= ( Y
= ( power_int @ real @ ( numeral_numeral @ real @ X3 ) @ N ) ) ) ) ).
% of_real_eq_numeral_power_int_cancel_iff
thf(fact_6731_numeral__power__int__eq__of__real__cancel__iff,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [X3: num,N: int,Y: real] :
( ( ( power_int @ A @ ( numeral_numeral @ A @ X3 ) @ N )
= ( real_Vector_of_real @ A @ Y ) )
= ( ( power_int @ real @ ( numeral_numeral @ real @ X3 ) @ N )
= Y ) ) ) ).
% numeral_power_int_eq_of_real_cancel_iff
thf(fact_6732_power__int__add__numeral2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M2: num,N: num,B2: A] :
( ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ M2 ) ) @ ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) @ B2 ) ) ) ).
% power_int_add_numeral2
thf(fact_6733_power__int__add__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M2: num,N: num] :
( ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ M2 ) ) @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ N ) ) )
= ( power_int @ A @ X3 @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% power_int_add_numeral
thf(fact_6734_power__int__mono__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ B2 @ N ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ) ).
% power_int_mono_iff
thf(fact_6735_power__int__minus__left__odd,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int,A3: A] :
( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% power_int_minus_left_odd
thf(fact_6736_power__int__minus__left__even,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int,A3: A] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_int @ A @ A3 @ N ) ) ) ) ).
% power_int_minus_left_even
thf(fact_6737_rev__ImageI,axiom,
! [B: $tType,A: $tType,A3: A,A6: set @ A,B2: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ A6 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R2 )
=> ( member @ B @ B2 @ ( image @ A @ B @ R2 @ A6 ) ) ) ) ).
% rev_ImageI
thf(fact_6738_Image__iff,axiom,
! [A: $tType,B: $tType,B2: A,R2: set @ ( product_prod @ B @ A ),A6: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R2 @ A6 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ B2 ) @ R2 ) ) ) ) ).
% Image_iff
thf(fact_6739_ImageE,axiom,
! [A: $tType,B: $tType,B2: A,R2: set @ ( product_prod @ B @ A ),A6: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R2 @ A6 ) )
=> ~ ! [X5: B] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X5 @ B2 ) @ R2 )
=> ~ ( member @ B @ X5 @ A6 ) ) ) ).
% ImageE
thf(fact_6740_Image__def,axiom,
! [B: $tType,A: $tType] :
( ( image @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B ),S7: set @ A] :
( collect @ B
@ ^ [Y3: B] :
? [X4: A] :
( ( member @ A @ X4 @ S7 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R5 ) ) ) ) ) ).
% Image_def
thf(fact_6741_Image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ B @ A ),B5: C > ( set @ B ),A6: set @ C] :
( ( image @ B @ A @ R2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B5 @ A6 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( image @ B @ A @ R2 @ ( B5 @ X4 ) )
@ A6 ) ) ) ).
% Image_UN
thf(fact_6742_finite__Image,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),A6: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( finite_finite2 @ B @ ( image @ A @ B @ R @ A6 ) ) ) ).
% finite_Image
thf(fact_6743_Image__Un,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),A6: set @ B,B5: set @ B] :
( ( image @ B @ A @ R @ ( sup_sup @ ( set @ B ) @ A6 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ R @ A6 ) @ ( image @ B @ A @ R @ B5 ) ) ) ).
% Image_Un
thf(fact_6744_Un__Image,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),S3: set @ ( product_prod @ B @ A ),A6: set @ B] :
( ( image @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ R @ S3 ) @ A6 )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ R @ A6 ) @ ( image @ B @ A @ S3 @ A6 ) ) ) ).
% Un_Image
thf(fact_6745_relcomp__Image,axiom,
! [A: $tType,C: $tType,B: $tType,X6: set @ ( product_prod @ B @ C ),Y8: set @ ( product_prod @ C @ A ),Z7: set @ B] :
( ( image @ B @ A @ ( relcomp @ B @ C @ A @ X6 @ Y8 ) @ Z7 )
= ( image @ C @ A @ Y8 @ ( image @ B @ C @ X6 @ Z7 ) ) ) ).
% relcomp_Image
thf(fact_6746_zero__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X3 @ N ) ) ) ) ).
% zero_le_power_int
thf(fact_6747_Image__mono,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),A11: set @ A,A6: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A11 @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R4 @ A11 ) @ ( image @ A @ B @ R2 @ A6 ) ) ) ) ).
% Image_mono
thf(fact_6748_Image__closed__trancl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ X6 ) @ X6 )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ X6 )
= X6 ) ) ).
% Image_closed_trancl
thf(fact_6749_Image__Int__subset,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),A6: set @ B,B5: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R @ ( inf_inf @ ( set @ B ) @ A6 @ B5 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ R @ A6 ) @ ( image @ B @ A @ R @ B5 ) ) ) ).
% Image_Int_subset
thf(fact_6750_power__int__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N5: int,A3: A] :
( ( ord_less_eq @ int @ N @ N5 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N5 ) ) ) ) ) ).
% power_int_increasing
thf(fact_6751_countable__Image,axiom,
! [B: $tType,A: $tType,Y8: set @ A,X6: set @ ( product_prod @ A @ B )] :
( ! [Y4: A] :
( ( member @ A @ Y4 @ Y8 )
=> ( countable_countable @ B @ ( image @ A @ B @ X6 @ ( insert2 @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( countable_countable @ A @ Y8 )
=> ( countable_countable @ B @ ( image @ A @ B @ X6 @ Y8 ) ) ) ) ).
% countable_Image
thf(fact_6752_wfE__pf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ( wf @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( image @ A @ A @ R @ A6 ) )
=> ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% wfE_pf
thf(fact_6753_wfI__pf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A10: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A10 @ ( image @ A @ A @ R @ A10 ) )
=> ( A10
= ( bot_bot @ ( set @ A ) ) ) )
=> ( wf @ A @ R ) ) ).
% wfI_pf
thf(fact_6754_equiv__class__self,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( member @ A @ A3 @ A6 )
=> ( member @ A @ A3 @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_self
thf(fact_6755_quotientI,axiom,
! [A: $tType,X3: A,A6: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X3 @ A6 )
=> ( member @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( equiv_quotient @ A @ A6 @ R2 ) ) ) ).
% quotientI
thf(fact_6756_quotientE,axiom,
! [A: $tType,X6: set @ A,A6: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R2 ) )
=> ~ ! [X5: A] :
( ( X6
= ( image @ A @ A @ R2 @ ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ~ ( member @ A @ X5 @ A6 ) ) ) ).
% quotientE
thf(fact_6757_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
@ ^ [B8: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B8 ) @ R2 ) ) ) ).
% Image_singleton
thf(fact_6758_proj__def,axiom,
! [A: $tType,B: $tType] :
( ( equiv_proj @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ A ),X4: B] : ( image @ B @ A @ R5 @ ( insert2 @ B @ X4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% proj_def
thf(fact_6759_Image__INT__subset,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ B @ A ),B5: C > ( set @ B ),A6: set @ C] :
( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B5 @ A6 ) ) )
@ ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( image @ B @ A @ R2 @ ( B5 @ X4 ) )
@ A6 ) ) ) ).
% Image_INT_subset
thf(fact_6760_power__int__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A,N: int] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( power_int @ A @ X3 @ N ) @ ( power_int @ A @ Y @ N ) ) ) ) ) ) ).
% power_int_mono
thf(fact_6761_one__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,N: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ X3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X3 @ N ) ) ) ) ) ).
% one_le_power_int
thf(fact_6762_equiv__class__eq__iff,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
= ( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( member @ A @ X3 @ A6 )
& ( member @ A @ Y @ A6 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_6763_eq__equiv__class__iff,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( member @ A @ Y @ A6 )
=> ( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_6764_equiv__class__eq,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_eq
thf(fact_6765_eq__equiv__class,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A,A6: set @ A] :
( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( member @ A @ B2 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 ) ) ) ) ).
% eq_equiv_class
thf(fact_6766_refines__equiv__class__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A ),A6: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S3 )
=> ( ( equiv_equiv @ A @ A6 @ R )
=> ( ( equiv_equiv @ A @ A6 @ S3 )
=> ( ( image @ A @ A @ R @ ( image @ A @ A @ S3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_6767_refines__equiv__class__eq2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A ),A6: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S3 )
=> ( ( equiv_equiv @ A @ A6 @ R )
=> ( ( equiv_equiv @ A @ A6 @ S3 )
=> ( ( image @ A @ A @ S3 @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_6768_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_6769_UN__Image,axiom,
! [A: $tType,B: $tType,C: $tType,X6: C > ( set @ ( product_prod @ B @ A ) ),I5: set @ C,S3: set @ B] :
( ( image @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ X6 @ I5 ) ) @ S3 )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [I4: C] : ( image @ B @ A @ ( X6 @ I4 ) @ S3 )
@ I5 ) ) ) ).
% UN_Image
thf(fact_6770_power__int__strict__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,N: int] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ B2 @ N ) ) ) ) ) ) ).
% power_int_strict_mono
thf(fact_6771_power__int__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,N: int] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ B2 @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_antimono
thf(fact_6772_power__int__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N5: int,A3: A] :
( ( ord_less_eq @ int @ N @ N5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( A3
!= ( zero_zero @ A ) )
| ( N5
!= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N5 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_6773_power__int__le__one,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ X3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ X3 @ N ) @ ( one_one @ A ) ) ) ) ) ) ).
% power_int_le_one
thf(fact_6774_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,M2: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ X3 @ M2 ) @ ( power_int @ A @ X3 @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ M2 @ N ) ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_6775_power__int__minus__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int,A3: A] :
( ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_int @ A @ A3 @ N ) ) )
& ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_minus_left
thf(fact_6776_equiv__class__subset,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_subset
thf(fact_6777_subset__equiv__class,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),B2: A,A3: A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ A @ B2 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 ) ) ) ) ).
% subset_equiv_class
thf(fact_6778_equiv__class__nondisjoint,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,A3: A,B2: A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_6779_singleton__quotient,axiom,
! [A: $tType,X3: A,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
= ( insert2 @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% singleton_quotient
thf(fact_6780_quotient__def,axiom,
! [A: $tType] :
( ( equiv_quotient @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A] : ( insert2 @ ( set @ A ) @ ( image @ A @ A @ R5 @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A7 ) ) ) ) ).
% quotient_def
thf(fact_6781_power__int__def,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ( ( power_int @ A )
= ( ^ [X4: A,N3: int] : ( if @ A @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N3 ) @ ( power_power @ A @ X4 @ ( nat2 @ N3 ) ) @ ( power_power @ A @ ( inverse_inverse @ A @ X4 ) @ ( nat2 @ ( uminus_uminus @ int @ N3 ) ) ) ) ) ) ) ).
% power_int_def
thf(fact_6782_Image__fold,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( ( image @ A @ B @ R @ S3 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X4: A,Y3: B,A7: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X4 @ S3 ) @ ( insert2 @ B @ Y3 @ A7 ) @ A7 ) )
@ ( bot_bot @ ( set @ B ) )
@ R ) ) ) ).
% Image_fold
thf(fact_6783_congruent2__implies__congruent__UN,axiom,
! [B: $tType,C: $tType,A: $tType,A19: set @ A,R1: set @ ( product_prod @ A @ A ),A25: set @ B,R22: set @ ( product_prod @ B @ B ),F3: A > B > ( set @ C ),A3: B] :
( ( equiv_equiv @ A @ A19 @ R1 )
=> ( ( equiv_equiv @ B @ A25 @ R22 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R22 @ F3 )
=> ( ( member @ B @ A3 @ A25 )
=> ( equiv_congruent @ A @ ( set @ C ) @ R1
@ ^ [X15: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F3 @ X15 ) @ ( image @ B @ B @ R22 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ) ).
% congruent2_implies_congruent_UN
thf(fact_6784_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A19: set @ A,R1: set @ ( product_prod @ A @ A ),A25: set @ B,R22: set @ ( product_prod @ B @ B ),F3: A > B > ( set @ C ),A1: A,A22: B] :
( ( equiv_equiv @ A @ A19 @ R1 )
=> ( ( equiv_equiv @ B @ A25 @ R22 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R22 @ F3 )
=> ( ( member @ A @ A1 @ A19 )
=> ( ( member @ B @ A22 @ A25 )
=> ( ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [X15: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F3 @ X15 ) @ ( image @ B @ B @ R22 @ ( insert2 @ B @ A22 @ ( bot_bot @ ( set @ B ) ) ) ) ) )
@ ( image @ A @ A @ R1 @ ( insert2 @ A @ A1 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F3 @ A1 @ A22 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_6785_congruent2I_H,axiom,
! [C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F3: A > B > C] :
( ! [Y15: A,Z1: A,Y23: B,Z22: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y15 @ Z1 ) @ R1 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y23 @ Z22 ) @ R22 )
=> ( ( F3 @ Y15 @ Y23 )
= ( F3 @ Z1 @ Z22 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F3 ) ) ).
% congruent2I'
thf(fact_6786_congruent2D,axiom,
! [A: $tType,C: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F3: A > B > C,Y1: A,Z12: A,Y2: B,Z23: B] :
( ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z12 ) @ R1 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y2 @ Z23 ) @ R22 )
=> ( ( F3 @ Y1 @ Y2 )
= ( F3 @ Z12 @ Z23 ) ) ) ) ) ).
% congruent2D
thf(fact_6787_congruent2I,axiom,
! [C: $tType,B: $tType,A: $tType,A19: set @ A,R1: set @ ( product_prod @ A @ A ),A25: set @ B,R22: set @ ( product_prod @ B @ B ),F3: A > B > C] :
( ( equiv_equiv @ A @ A19 @ R1 )
=> ( ( equiv_equiv @ B @ A25 @ R22 )
=> ( ! [Y4: A,Z3: A,W2: B] :
( ( member @ B @ W2 @ A25 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R1 )
=> ( ( F3 @ Y4 @ W2 )
= ( F3 @ Z3 @ W2 ) ) ) )
=> ( ! [Y4: B,Z3: B,W2: A] :
( ( member @ A @ W2 @ A19 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y4 @ Z3 ) @ R22 )
=> ( ( F3 @ W2 @ Y4 )
= ( F3 @ W2 @ Z3 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F3 ) ) ) ) ) ).
% congruent2I
thf(fact_6788_congruent2__commuteI,axiom,
! [B: $tType,A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > A > B] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ! [Y4: A,Z3: A] :
( ( member @ A @ Y4 @ A6 )
=> ( ( member @ A @ Z3 @ A6 )
=> ( ( F3 @ Y4 @ Z3 )
= ( F3 @ Z3 @ Y4 ) ) ) )
=> ( ! [Y4: A,Z3: A,W2: A] :
( ( member @ A @ W2 @ A6 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 )
=> ( ( F3 @ W2 @ Y4 )
= ( F3 @ W2 @ Z3 ) ) ) )
=> ( equiv_congruent2 @ A @ A @ B @ R2 @ R2 @ F3 ) ) ) ) ).
% congruent2_commuteI
thf(fact_6789_listrel__Cons,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),X3: B,Xs2: list @ B] :
( ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R2 ) @ ( insert2 @ ( list @ B ) @ ( cons @ B @ X3 @ Xs2 ) @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) )
= ( set_Cons @ A @ ( image @ B @ A @ R2 @ ( insert2 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R2 ) @ ( insert2 @ ( list @ B ) @ Xs2 @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) ) ) ) ).
% listrel_Cons
thf(fact_6790_subset__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B2 @ ( 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 @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_6791_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_6792_subset__Image__Image__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A,B5: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ A6 ) @ ( image @ A @ A @ R2 @ B5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ B5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R2 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_6793_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_6794_DeMoivre2,axiom,
! [R2: real,A3: real,N: nat] :
( ( power_power @ complex @ ( rcis @ R2 @ A3 ) @ N )
= ( rcis @ ( power_power @ real @ R2 @ N ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A3 ) ) ) ).
% DeMoivre2
thf(fact_6795_in__listsI,axiom,
! [A: $tType,Xs2: list @ A,A6: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( member @ A @ X5 @ A6 ) )
=> ( member @ ( list @ A ) @ Xs2 @ ( lists @ A @ A6 ) ) ) ).
% in_listsI
thf(fact_6796_in__listsD,axiom,
! [A: $tType,Xs2: list @ A,A6: set @ A] :
( ( member @ ( list @ A ) @ Xs2 @ ( lists @ A @ A6 ) )
=> ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( member @ A @ X @ A6 ) ) ) ).
% in_listsD
thf(fact_6797_in__lists__conv__set,axiom,
! [A: $tType,Xs2: list @ A,A6: set @ A] :
( ( member @ ( list @ A ) @ Xs2 @ ( lists @ A @ A6 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( member @ A @ X4 @ A6 ) ) ) ) ).
% in_lists_conv_set
thf(fact_6798_lists__mono,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( lists @ A @ A6 ) @ ( lists @ A @ B5 ) ) ) ).
% lists_mono
thf(fact_6799_lists__eq__set,axiom,
! [A: $tType] :
( ( lists @ A )
= ( ^ [A7: set @ A] :
( collect @ ( list @ A )
@ ^ [Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A7 ) ) ) ) ).
% lists_eq_set
thf(fact_6800_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_6801_Collect__finite__subset__eq__lists,axiom,
! [A: $tType,T4: set @ A] :
( ( collect @ ( set @ A )
@ ^ [A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
& ( ord_less_eq @ ( set @ A ) @ A7 @ T4 ) ) )
= ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( lists @ A @ T4 ) ) ) ).
% Collect_finite_subset_eq_lists
thf(fact_6802_lexn_Osimps_I2_J,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),N: nat] :
( ( lexn @ A @ R2 @ ( suc @ N ) )
= ( inf_inf @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image2 @ ( product_prod @ ( product_prod @ A @ ( list @ A ) ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_map_prod @ ( product_prod @ A @ ( list @ A ) ) @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( list @ A ) @ ( product_case_prod @ A @ ( list @ A ) @ ( list @ A ) @ ( cons @ A ) ) @ ( product_case_prod @ A @ ( list @ A ) @ ( list @ A ) @ ( cons @ A ) ) ) @ ( lex_prod @ A @ ( list @ A ) @ R2 @ ( lexn @ A @ R2 @ N ) ) )
@ ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= ( suc @ N ) ) ) ) ) ) ) ).
% lexn.simps(2)
thf(fact_6803_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F3: nat > A > A,A3: nat,B2: 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 @ B2 @ Acc3 ) ) ) )
=> ( ( ( ord_less @ nat @ B2 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F3 @ A3 @ B2 @ Acc3 )
= Acc3 ) )
& ( ~ ( ord_less @ nat @ B2 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F3 @ A3 @ B2 @ Acc3 )
= ( set_fo6178422350223883121st_nat @ A @ F3 @ ( plus_plus @ nat @ A3 @ ( one_one @ nat ) ) @ B2 @ ( F3 @ A3 @ Acc3 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_6804_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X4: A] : X4
@ ^ [Y3: B] : Y3 )
= ( ^ [Z4: product_prod @ A @ B] : Z4 ) ) ).
% map_prod_ident
thf(fact_6805_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G3: D > B,A3: C,B2: D] :
( ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ ( product_Pair @ C @ D @ A3 @ B2 ) )
= ( product_Pair @ A @ B @ ( F3 @ A3 ) @ ( G3 @ B2 ) ) ) ).
% map_prod_simp
thf(fact_6806_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > A,G3: D > B,X3: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ X3 ) )
= ( F3 @ ( product_fst @ C @ D @ X3 ) ) ) ).
% fst_map_prod
thf(fact_6807_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > B,G3: D > A,X3: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F3 @ G3 @ X3 ) )
= ( G3 @ ( product_snd @ C @ D @ X3 ) ) ) ).
% snd_map_prod
thf(fact_6808_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B ),F3: A > C,G3: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F3 @ A3 ) @ ( G3 @ B2 ) ) @ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F3 @ G3 ) @ R ) ) ) ).
% map_prod_imageI
thf(fact_6809_fst__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F3: A > C,G3: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F3 @ G3 ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F3 @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_map_prod
thf(fact_6810_snd__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F3: A > D,G3: B > C] :
( ( comp @ ( product_prod @ D @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ D @ C ) @ ( product_map_prod @ A @ D @ B @ C @ F3 @ G3 ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ G3 @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_map_prod
thf(fact_6811_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F3: C > A,G3: D > B,R: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 ) @ R ) )
=> ~ ! [X5: C,Y4: D] :
( ( C3
= ( product_Pair @ A @ B @ ( F3 @ X5 ) @ ( G3 @ Y4 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X5 @ Y4 ) @ R ) ) ) ).
% prod_fun_imageE
thf(fact_6812_map__prod__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( product_map_prod @ A @ C @ B @ D )
= ( ^ [F4: A > C,G4: B > D] :
( product_case_prod @ A @ B @ ( product_prod @ C @ D )
@ ^ [X4: A,Y3: B] : ( product_Pair @ C @ D @ ( F4 @ X4 ) @ ( G4 @ Y3 ) ) ) ) ) ).
% map_prod_def
thf(fact_6813_map__prod_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F: $tType,D: $tType,B: $tType,F3: C > E,G3: D > F,H: A > C,I: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F @ F3 @ G3 ) @ ( product_map_prod @ A @ C @ B @ D @ H @ I ) )
= ( product_map_prod @ A @ E @ B @ F @ ( comp @ C @ E @ A @ F3 @ H ) @ ( comp @ D @ F @ B @ G3 @ I ) ) ) ).
% map_prod.comp
thf(fact_6814_map__prod_Ocompositionality,axiom,
! [D: $tType,F: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F3: C > E,G3: D > F,H: A > C,I: B > D,Prod: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F @ F3 @ G3 @ ( product_map_prod @ A @ C @ B @ D @ H @ I @ Prod ) )
= ( product_map_prod @ A @ E @ B @ F @ ( comp @ C @ E @ A @ F3 @ H ) @ ( comp @ D @ F @ B @ G3 @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_6815_map__prod__compose,axiom,
! [D: $tType,C: $tType,A: $tType,E: $tType,F: $tType,B: $tType,F1: E > C,F22: A > E,G1: F > D,G22: B > F] :
( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F @ D @ B @ G1 @ G22 ) )
= ( comp @ ( product_prod @ E @ F ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_6816_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F3: A > B,G3: C > D] :
( ( ( image2 @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image2 @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image2 @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F3 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_6817_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: nat > A > A,A1: nat,A22: nat,A33: A,P: ( nat > A > A ) > nat > nat > A > $o] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ A0 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A1 @ ( product_Pair @ nat @ A @ A22 @ A33 ) ) ) )
=> ( ! [F2: nat > A > A,A5: nat,B4: nat,Acc: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A5 @ ( product_Pair @ nat @ A @ B4 @ Acc ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B4 @ A5 )
=> ( P @ F2 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B4 @ ( F2 @ A5 @ Acc ) ) )
=> ( P @ F2 @ A5 @ B4 @ Acc ) ) )
=> ( P @ A0 @ A1 @ A22 @ A33 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_6818_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X3: nat > A > A,Xa2: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X3 @ Xa2 @ Xb @ 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 ) ) @ X3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa2 @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less @ nat @ Xb @ Xa2 )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa2 )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X3 @ ( plus_plus @ nat @ Xa2 @ ( one_one @ nat ) ) @ Xb @ ( X3 @ Xa2 @ Xc ) ) ) ) )
=> ~ ( 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 ) ) @ X3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa2 @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_6819_tendsto__iff__uniformity,axiom,
! [A: $tType,B: $tType] :
( ( topolo7287701948861334536_space @ B )
=> ! [F3: A > B,L: B,F6: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F6 )
= ( ! [E6: ( product_prod @ B @ B ) > $o] :
( ( eventually @ ( product_prod @ B @ B ) @ E6 @ ( topolo7806501430040627800ormity @ B ) )
=> ( eventually @ A
@ ^ [X4: A] : ( E6 @ ( product_Pair @ B @ B @ ( F3 @ X4 ) @ L ) )
@ F6 ) ) ) ) ) ).
% tendsto_iff_uniformity
thf(fact_6820_independent__explicit__finite__subsets,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A6: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ A6 ) )
= ( ! [S6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S6 @ A6 )
=> ( ( finite_finite2 @ A @ S6 )
=> ! [U2: A > real] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V5 ) @ V5 )
@ S6 )
= ( zero_zero @ A ) )
=> ! [X4: A] :
( ( member @ A @ X4 @ S6 )
=> ( ( U2 @ X4 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% independent_explicit_finite_subsets
thf(fact_6821_independent__empty,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ~ ( real_V358717886546972837endent @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% independent_empty
thf(fact_6822_dependent__single,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A] :
( ( real_V358717886546972837endent @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% dependent_single
thf(fact_6823_independent__Union__directed,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [C4: set @ ( set @ A )] :
( ! [C2: set @ A,D2: set @ A] :
( ( member @ ( set @ A ) @ C2 @ C4 )
=> ( ( member @ ( set @ A ) @ D2 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ C2 @ D2 )
| ( ord_less_eq @ ( set @ A ) @ D2 @ C2 ) ) ) )
=> ( ! [C2: set @ A] :
( ( member @ ( set @ A ) @ C2 @ C4 )
=> ~ ( real_V358717886546972837endent @ A @ C2 ) )
=> ~ ( real_V358717886546972837endent @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) ) ) ) ) ).
% independent_Union_directed
thf(fact_6824_dependent__mono,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B5: set @ A,A6: set @ A] :
( ( real_V358717886546972837endent @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( real_V358717886546972837endent @ A @ A6 ) ) ) ) ).
% dependent_mono
thf(fact_6825_independent__mono,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ~ ( real_V358717886546972837endent @ A @ B5 ) ) ) ) ).
% independent_mono
thf(fact_6826_uniformity__transE,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [E5: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) )
=> ~ ! [D8: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ D8 @ ( topolo7806501430040627800ormity @ A ) )
=> ~ ! [X: A,Y6: A] :
( ( D8 @ ( product_Pair @ A @ A @ X @ Y6 ) )
=> ! [Z5: A] :
( ( D8 @ ( product_Pair @ A @ A @ Y6 @ Z5 ) )
=> ( E5 @ ( product_Pair @ A @ A @ X @ Z5 ) ) ) ) ) ) ) ).
% uniformity_transE
thf(fact_6827_uniformity__trans,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [E5: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) )
=> ? [D8: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ D8 @ ( topolo7806501430040627800ormity @ A ) )
& ! [X: A,Y6: A,Z5: A] :
( ( D8 @ ( product_Pair @ A @ A @ X @ Y6 ) )
=> ( ( D8 @ ( product_Pair @ A @ A @ Y6 @ Z5 ) )
=> ( E5 @ ( product_Pair @ A @ A @ X @ Z5 ) ) ) ) ) ) ) ).
% uniformity_trans
thf(fact_6828_uniformity__refl,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [E5: ( product_prod @ A @ A ) > $o,X3: A] :
( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) )
=> ( E5 @ ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% uniformity_refl
thf(fact_6829_uniformity__sym,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [E5: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) )
=> ( eventually @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y3: A] : ( E5 @ ( product_Pair @ A @ A @ Y3 @ X4 ) ) )
@ ( topolo7806501430040627800ormity @ A ) ) ) ) ).
% uniformity_sym
thf(fact_6830_Cauchy__uniform__iff,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [P4: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ P4 @ ( topolo7806501430040627800ormity @ A ) )
=> ? [N6: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ! [M5: nat] :
( ( ord_less_eq @ nat @ N6 @ M5 )
=> ( P4 @ ( product_Pair @ A @ A @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) ) ) ) ) ) ) ) ).
% Cauchy_uniform_iff
thf(fact_6831_totally__bounded__def,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo6688025880775521714ounded @ A )
= ( ^ [S6: set @ A] :
! [E6: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ E6 @ ( topolo7806501430040627800ormity @ A ) )
=> ? [X8: set @ A] :
( ( finite_finite2 @ A @ X8 )
& ! [X4: A] :
( ( member @ A @ X4 @ S6 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ X8 )
& ( E6 @ ( product_Pair @ A @ A @ Y3 @ X4 ) ) ) ) ) ) ) ) ) ).
% totally_bounded_def
thf(fact_6832_independentD__unique,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B5: set @ A,X6: A > real,Y8: A > real] :
( ~ ( real_V358717886546972837endent @ A @ B5 )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( X6 @ X4 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( X6 @ X4 )
!= ( zero_zero @ real ) ) )
@ B5 )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( Y8 @ X4 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( Y8 @ X4 )
!= ( zero_zero @ real ) ) )
@ B5 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( X6 @ X4 ) @ X4 )
@ ( collect @ A
@ ^ [X4: A] :
( ( X6 @ X4 )
!= ( zero_zero @ real ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( Y8 @ X4 ) @ X4 )
@ ( collect @ A
@ ^ [X4: A] :
( ( Y8 @ X4 )
!= ( zero_zero @ real ) ) ) ) )
=> ( X6 = Y8 ) ) ) ) ) ) ) ) ).
% independentD_unique
thf(fact_6833_eventually__uniformity__metric,axiom,
! [A: $tType] :
( ( real_V768167426530841204y_dist @ A )
=> ! [P: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ P @ ( topolo7806501430040627800ormity @ A ) )
= ( ? [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
& ! [X4: A,Y3: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ Y3 ) @ E4 )
=> ( P @ ( product_Pair @ A @ A @ X4 @ Y3 ) ) ) ) ) ) ) ).
% eventually_uniformity_metric
thf(fact_6834_independentD,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A,T2: set @ A,U: A > real,V2: A] :
( ~ ( real_V358717886546972837endent @ A @ S )
=> ( ( finite_finite2 @ A @ T2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( U @ V5 ) @ V5 )
@ T2 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ V2 @ T2 )
=> ( ( U @ V2 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% independentD
thf(fact_6835_dependent__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V358717886546972837endent @ A )
= ( ^ [B6: set @ A] :
? [X8: A > real] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( X8 @ X4 )
!= ( zero_zero @ real ) ) ) )
& ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( X8 @ X4 )
!= ( zero_zero @ real ) ) )
@ B6 )
& ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( X8 @ X4 ) @ X4 )
@ ( collect @ A
@ ^ [X4: A] :
( ( X8 @ X4 )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
& ? [X4: A] :
( ( X8 @ X4 )
!= ( zero_zero @ real ) ) ) ) ) ) ).
% dependent_alt
thf(fact_6836_independent__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B5: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ B5 ) )
= ( ! [X8: A > real] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( X8 @ X4 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( X8 @ X4 )
!= ( zero_zero @ real ) ) )
@ B5 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( X8 @ X4 ) @ X4 )
@ ( collect @ A
@ ^ [X4: A] :
( ( X8 @ X4 )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
=> ! [X4: A] :
( ( X8 @ X4 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% independent_alt
thf(fact_6837_independentD__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B5: set @ A,X6: A > real,X3: A] :
( ~ ( real_V358717886546972837endent @ A @ B5 )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( X6 @ X4 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( X6 @ X4 )
!= ( zero_zero @ real ) ) )
@ B5 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( X6 @ X4 ) @ X4 )
@ ( collect @ A
@ ^ [X4: A] :
( ( X6 @ X4 )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
=> ( ( X6 @ X3 )
= ( zero_zero @ real ) ) ) ) ) ) ) ).
% independentD_alt
thf(fact_6838_dependent__explicit,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V358717886546972837endent @ A )
= ( ^ [S7: set @ A] :
? [T3: set @ A] :
( ( finite_finite2 @ A @ T3 )
& ( ord_less_eq @ ( set @ A ) @ T3 @ S7 )
& ? [U2: A > real] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V5 ) @ V5 )
@ T3 )
= ( zero_zero @ A ) )
& ? [X4: A] :
( ( member @ A @ X4 @ T3 )
& ( ( U2 @ X4 )
!= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% dependent_explicit
thf(fact_6839_independent__explicit__module,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ S ) )
= ( ! [T3: set @ A,U2: A > real,V5: A] :
( ( finite_finite2 @ A @ T3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T3 @ S )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [W3: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ W3 ) @ W3 )
@ T3 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ V5 @ T3 )
=> ( ( U2 @ V5 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% independent_explicit_module
thf(fact_6840_divmod__nat__code,axiom,
( divmod_nat
= ( ^ [M5: nat,N3: nat] :
( product_map_prod @ code_integer @ nat @ code_integer @ nat @ code_nat_of_integer @ code_nat_of_integer
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( ( code_integer_of_nat @ M5 )
= ( 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 )
@ ( ( code_integer_of_nat @ N3 )
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( code_integer_of_nat @ M5 ) )
@ ( code_divmod_abs @ ( code_integer_of_nat @ M5 ) @ ( code_integer_of_nat @ N3 ) ) ) ) ) ) ) ).
% divmod_nat_code
thf(fact_6841_uniformity__trans_H,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [E5: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) )
=> ( eventually @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) )
@ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o
@ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o )
@ ^ [X4: A,Y3: A] :
( product_case_prod @ A @ A @ $o
@ ^ [Y7: A,Z4: A] :
( ( Y3 = Y7 )
=> ( E5 @ ( product_Pair @ A @ A @ X4 @ Z4 ) ) ) ) ) )
@ ( prod_filter @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ ( topolo7806501430040627800ormity @ A ) @ ( topolo7806501430040627800ormity @ A ) ) ) ) ) ).
% uniformity_trans'
thf(fact_6842_prod__filter__eq__bot,axiom,
! [A: $tType,B: $tType,A6: filter @ A,B5: filter @ B] :
( ( ( prod_filter @ A @ B @ A6 @ B5 )
= ( bot_bot @ ( filter @ ( product_prod @ A @ B ) ) ) )
= ( ( A6
= ( bot_bot @ ( filter @ A ) ) )
| ( B5
= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% prod_filter_eq_bot
thf(fact_6843_prod__filter__mono,axiom,
! [A: $tType,B: $tType,F6: filter @ A,F11: filter @ A,G7: filter @ B,G8: filter @ B] :
( ( ord_less_eq @ ( filter @ A ) @ F6 @ F11 )
=> ( ( ord_less_eq @ ( filter @ B ) @ G7 @ G8 )
=> ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ ( prod_filter @ A @ B @ F6 @ G7 ) @ ( prod_filter @ A @ B @ F11 @ G8 ) ) ) ) ).
% prod_filter_mono
thf(fact_6844_prod__filter__mono__iff,axiom,
! [A: $tType,B: $tType,A6: filter @ A,B5: filter @ B,C4: filter @ A,D4: filter @ B] :
( ( A6
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( B5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ ( prod_filter @ A @ B @ A6 @ B5 ) @ ( prod_filter @ A @ B @ C4 @ D4 ) )
= ( ( ord_less_eq @ ( filter @ A ) @ A6 @ C4 )
& ( ord_less_eq @ ( filter @ B ) @ B5 @ D4 ) ) ) ) ) ).
% prod_filter_mono_iff
thf(fact_6845_prod__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( prod_filter @ A @ B )
= ( ^ [F9: filter @ A,G9: 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 ) )
@ ^ [P4: A > $o,Q6: B > $o] :
( principal @ ( product_prod @ A @ B )
@ ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y3: B] :
( ( P4 @ X4 )
& ( Q6 @ Y3 ) ) ) ) ) )
@ ( collect @ ( product_prod @ ( A > $o ) @ ( B > $o ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ $o
@ ^ [P4: A > $o,Q6: B > $o] :
( ( eventually @ A @ P4 @ F9 )
& ( eventually @ B @ Q6 @ G9 ) ) ) ) ) ) ) ) ).
% prod_filter_def
thf(fact_6846_eventually__prod__filter,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,F6: filter @ A,G7: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ P @ ( prod_filter @ A @ B @ F6 @ G7 ) )
= ( ? [Pf: A > $o,Pg: B > $o] :
( ( eventually @ A @ Pf @ F6 )
& ( eventually @ B @ Pg @ G7 )
& ! [X4: A,Y3: B] :
( ( Pf @ X4 )
=> ( ( Pg @ Y3 )
=> ( P @ ( product_Pair @ A @ B @ X4 @ Y3 ) ) ) ) ) ) ) ).
% eventually_prod_filter
thf(fact_6847_eventually__prod__same,axiom,
! [A: $tType,P: ( product_prod @ A @ A ) > $o,F6: filter @ A] :
( ( eventually @ ( product_prod @ A @ A ) @ P @ ( prod_filter @ A @ A @ F6 @ F6 ) )
= ( ? [Q6: A > $o] :
( ( eventually @ A @ Q6 @ F6 )
& ! [X4: A,Y3: A] :
( ( Q6 @ X4 )
=> ( ( Q6 @ Y3 )
=> ( P @ ( product_Pair @ A @ A @ X4 @ Y3 ) ) ) ) ) ) ) ).
% eventually_prod_same
thf(fact_6848_integer__of__nat__numeral,axiom,
! [N: num] :
( ( code_integer_of_nat @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ code_integer @ N ) ) ).
% integer_of_nat_numeral
thf(fact_6849_prod__filter__INF2,axiom,
! [B: $tType,C: $tType,A: $tType,J5: set @ A,A6: filter @ B,B5: A > ( filter @ C )] :
( ( J5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ A6 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ B5 @ J5 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I4: A] : ( prod_filter @ B @ C @ A6 @ ( B5 @ I4 ) )
@ J5 ) ) ) ) ).
% prod_filter_INF2
thf(fact_6850_prod__filter__INF1,axiom,
! [B: $tType,C: $tType,A: $tType,I5: set @ A,A6: A > ( filter @ B ),B5: filter @ C] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ A6 @ I5 ) ) @ B5 )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I4: A] : ( prod_filter @ B @ C @ ( A6 @ I4 ) @ B5 )
@ I5 ) ) ) ) ).
% prod_filter_INF1
thf(fact_6851_prod__filter__INF,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,I5: set @ A,J5: set @ B,A6: A > ( filter @ C ),B5: 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 ) @ A6 @ I5 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ B5 @ 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 @ ( A6 @ I4 ) @ ( B5 @ J3 ) )
@ J5 ) )
@ I5 ) ) ) ) ) ).
% prod_filter_INF
thf(fact_6852_nhds__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A3: A,B2: B] :
( ( topolo7230453075368039082e_nhds @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) )
= ( prod_filter @ A @ B @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( topolo7230453075368039082e_nhds @ B @ B2 ) ) ) ) ).
% nhds_prod
thf(fact_6853_filterlim__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B,G7: filter @ B,F6: filter @ A,G3: A > C,H6: filter @ C] :
( ( filterlim @ A @ B @ F3 @ G7 @ F6 )
=> ( ( filterlim @ A @ C @ G3 @ H6 @ F6 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X4: A] : ( product_Pair @ B @ C @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( prod_filter @ B @ C @ G7 @ H6 )
@ F6 ) ) ) ).
% filterlim_Pair
thf(fact_6854_tendsto__add__Pair,axiom,
! [A: $tType] :
( ( topolo6943815403480290642id_add @ A )
=> ! [A3: A,B2: A] :
( filterlim @ ( product_prod @ A @ A ) @ A
@ ^ [X4: product_prod @ A @ A] : ( plus_plus @ A @ ( product_fst @ A @ A @ X4 ) @ ( product_snd @ A @ A @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A3 @ B2 ) )
@ ( prod_filter @ A @ A @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( topolo7230453075368039082e_nhds @ A @ B2 ) ) ) ) ).
% tendsto_add_Pair
thf(fact_6855_eventually__prodI,axiom,
! [A: $tType,B: $tType,P: A > $o,F6: filter @ A,Q: B > $o,G7: filter @ B] :
( ( eventually @ A @ P @ F6 )
=> ( ( eventually @ B @ Q @ G7 )
=> ( eventually @ ( product_prod @ A @ B )
@ ^ [X4: product_prod @ A @ B] :
( ( P @ ( product_fst @ A @ B @ X4 ) )
& ( Q @ ( product_snd @ A @ B @ X4 ) ) )
@ ( prod_filter @ A @ B @ F6 @ G7 ) ) ) ) ).
% eventually_prodI
thf(fact_6856_eventually__prod1,axiom,
! [A: $tType,B: $tType,B5: filter @ A,P: B > $o,A6: filter @ B] :
( ( B5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ^ [X4: B,Y3: A] : ( P @ X4 ) )
@ ( prod_filter @ B @ A @ A6 @ B5 ) )
= ( eventually @ B @ P @ A6 ) ) ) ).
% eventually_prod1
thf(fact_6857_eventually__prod2,axiom,
! [A: $tType,B: $tType,A6: filter @ A,P: B > $o,B5: filter @ B] :
( ( A6
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A] : P )
@ ( prod_filter @ A @ B @ A6 @ B5 ) )
= ( eventually @ B @ P @ B5 ) ) ) ).
% eventually_prod2
thf(fact_6858_prod__filter__assoc,axiom,
! [A: $tType,B: $tType,C: $tType,F6: filter @ A,G7: filter @ B,H6: filter @ C] :
( ( prod_filter @ ( product_prod @ A @ B ) @ C @ ( prod_filter @ A @ B @ F6 @ G7 ) @ H6 )
= ( 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 )
@ ^ [X4: 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 @ X4 @ Y3 ) ) ) )
@ ( prod_filter @ A @ ( product_prod @ B @ C ) @ F6 @ ( prod_filter @ B @ C @ G7 @ H6 ) ) ) ) ).
% prod_filter_assoc
thf(fact_6859_uniformly__continuous__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo7287701948861334536_space @ A )
& ( topolo7287701948861334536_space @ B ) )
=> ! [S: set @ A,F3: A > B,E5: ( product_prod @ B @ B ) > $o] :
( ( topolo6026614971017936543ous_on @ A @ B @ S @ F3 )
=> ( ( eventually @ ( product_prod @ B @ B ) @ E5 @ ( topolo7806501430040627800ormity @ B ) )
=> ( eventually @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y3: A] :
( ( member @ A @ X4 @ S )
=> ( ( member @ A @ Y3 @ S )
=> ( E5 @ ( product_Pair @ B @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) ) ) ) )
@ ( topolo7806501430040627800ormity @ A ) ) ) ) ) ).
% uniformly_continuous_onD
thf(fact_6860_filtermap__id_H,axiom,
! [A: $tType] :
( ( filtermap @ A @ A
@ ^ [X4: A] : X4 )
= ( ^ [F9: filter @ A] : F9 ) ) ).
% filtermap_id'
thf(fact_6861_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_6862_filtermap__principal,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B] :
( ( filtermap @ B @ A @ F3 @ ( principal @ B @ A6 ) )
= ( principal @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ).
% filtermap_principal
thf(fact_6863_filtermap__fun__inverse,axiom,
! [B: $tType,A: $tType,G3: A > B,F6: filter @ B,G7: filter @ A,F3: B > A] :
( ( filterlim @ A @ B @ G3 @ F6 @ G7 )
=> ( ( filterlim @ B @ A @ F3 @ G7 @ F6 )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( F3 @ ( G3 @ X4 ) )
= X4 )
@ G7 )
=> ( ( filtermap @ B @ A @ F3 @ F6 )
= G7 ) ) ) ) ).
% filtermap_fun_inverse
thf(fact_6864_eventually__filtermap,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: B > A,F6: filter @ B] :
( ( eventually @ A @ P @ ( filtermap @ B @ A @ F3 @ F6 ) )
= ( eventually @ B
@ ^ [X4: B] : ( P @ ( F3 @ X4 ) )
@ F6 ) ) ).
% eventually_filtermap
thf(fact_6865_prod__filtermap2,axiom,
! [B: $tType,A: $tType,C: $tType,F6: filter @ A,G3: C > B,G7: filter @ C] :
( ( prod_filter @ A @ B @ F6 @ ( filtermap @ C @ B @ G3 @ G7 ) )
= ( filtermap @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B ) @ ( product_apsnd @ C @ B @ A @ G3 ) @ ( prod_filter @ A @ C @ F6 @ G7 ) ) ) ).
% prod_filtermap2
thf(fact_6866_prod__filtermap1,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,F6: filter @ C,G7: filter @ B] :
( ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F3 @ F6 ) @ G7 )
= ( filtermap @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B ) @ ( product_apfst @ C @ A @ B @ F3 ) @ ( prod_filter @ C @ B @ F6 @ G7 ) ) ) ).
% prod_filtermap1
thf(fact_6867_filtermap__Pair,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > A,G3: C > B,F6: filter @ C] :
( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) )
@ ( filtermap @ C @ ( product_prod @ A @ B )
@ ^ [X4: C] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ F6 )
@ ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F3 @ F6 ) @ ( filtermap @ C @ B @ G3 @ F6 ) ) ) ).
% filtermap_Pair
thf(fact_6868_filtermap__inf,axiom,
! [A: $tType,B: $tType,F3: B > A,F13: filter @ B,F24: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ ( inf_inf @ ( filter @ B ) @ F13 @ F24 ) ) @ ( inf_inf @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ F13 ) @ ( filtermap @ B @ A @ F3 @ F24 ) ) ) ).
% filtermap_inf
thf(fact_6869_filtermap__mono,axiom,
! [B: $tType,A: $tType,F6: filter @ A,F11: filter @ A,F3: A > B] :
( ( ord_less_eq @ ( filter @ A ) @ F6 @ F11 )
=> ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ A @ B @ F3 @ F6 ) @ ( filtermap @ A @ B @ F3 @ F11 ) ) ) ).
% filtermap_mono
thf(fact_6870_filterlim__def,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F4: A > B,F26: filter @ B,F15: filter @ A] : ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ A @ B @ F4 @ F15 ) @ F26 ) ) ) ).
% filterlim_def
thf(fact_6871_filtermap__le__iff__le__filtercomap,axiom,
! [B: $tType,A: $tType,F3: B > A,F6: filter @ B,G7: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ F6 ) @ G7 )
= ( ord_less_eq @ ( filter @ B ) @ F6 @ ( filtercomap @ B @ A @ F3 @ G7 ) ) ) ).
% filtermap_le_iff_le_filtercomap
thf(fact_6872_filtermap__filtercomap,axiom,
! [B: $tType,A: $tType,F3: B > A,F6: filter @ A] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ ( filtercomap @ B @ A @ F3 @ F6 ) ) @ F6 ) ).
% filtermap_filtercomap
thf(fact_6873_filtercomap__filtermap,axiom,
! [B: $tType,A: $tType,F6: filter @ A,F3: A > B] : ( ord_less_eq @ ( filter @ A ) @ F6 @ ( filtercomap @ A @ B @ F3 @ ( filtermap @ A @ B @ F3 @ F6 ) ) ) ).
% filtercomap_filtermap
thf(fact_6874_filtermap__bot__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,F6: filter @ B] :
( ( ( filtermap @ B @ A @ F3 @ F6 )
= ( bot_bot @ ( filter @ A ) ) )
= ( F6
= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtermap_bot_iff
thf(fact_6875_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_6876_filtermap__sup,axiom,
! [A: $tType,B: $tType,F3: B > A,F13: filter @ B,F24: filter @ B] :
( ( filtermap @ B @ A @ F3 @ ( sup_sup @ ( filter @ B ) @ F13 @ F24 ) )
= ( sup_sup @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ F13 ) @ ( filtermap @ B @ A @ F3 @ F24 ) ) ) ).
% filtermap_sup
thf(fact_6877_filtermap__filtermap,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,G3: C > B,F6: filter @ C] :
( ( filtermap @ B @ A @ F3 @ ( filtermap @ C @ B @ G3 @ F6 ) )
= ( filtermap @ C @ A
@ ^ [X4: C] : ( F3 @ ( G3 @ X4 ) )
@ F6 ) ) ).
% filtermap_filtermap
thf(fact_6878_filtermap__ident,axiom,
! [A: $tType,F6: filter @ A] :
( ( filtermap @ A @ A
@ ^ [X4: A] : X4
@ F6 )
= F6 ) ).
% filtermap_ident
thf(fact_6879_filterlim__filtermap,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > B,F13: filter @ B,G3: C > A,F24: filter @ C] :
( ( filterlim @ A @ B @ F3 @ F13 @ ( filtermap @ C @ A @ G3 @ F24 ) )
= ( filterlim @ C @ B
@ ^ [X4: C] : ( F3 @ ( G3 @ X4 ) )
@ F13
@ F24 ) ) ).
% filterlim_filtermap
thf(fact_6880_filtermap__eq__strong,axiom,
! [B: $tType,A: $tType,F3: A > B,F6: filter @ A,G7: filter @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( filtermap @ A @ B @ F3 @ F6 )
= ( filtermap @ A @ B @ F3 @ G7 ) )
= ( F6 = G7 ) ) ) ).
% filtermap_eq_strong
thf(fact_6881_filtermap__SUP,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,F6: C > ( filter @ B ),B5: set @ C] :
( ( filtermap @ B @ A @ F3 @ ( complete_Sup_Sup @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F6 @ B5 ) ) )
= ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B8: C] : ( filtermap @ B @ A @ F3 @ ( F6 @ B8 ) )
@ B5 ) ) ) ).
% filtermap_SUP
thf(fact_6882_filtermap__def,axiom,
! [B: $tType,A: $tType] :
( ( filtermap @ A @ B )
= ( ^ [F4: A > B,F9: filter @ A] :
( abs_filter @ B
@ ^ [P4: B > $o] :
( eventually @ A
@ ^ [X4: A] : ( P4 @ ( F4 @ X4 ) )
@ F9 ) ) ) ) ).
% filtermap_def
thf(fact_6883_filtermap__mono__strong,axiom,
! [B: $tType,A: $tType,F3: A > B,F6: filter @ A,G7: filter @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ A @ B @ F3 @ F6 ) @ ( filtermap @ A @ B @ F3 @ G7 ) )
= ( ord_less_eq @ ( filter @ A ) @ F6 @ G7 ) ) ) ).
% filtermap_mono_strong
thf(fact_6884_filtermap__fst__prod__filter,axiom,
! [B: $tType,A: $tType,A6: filter @ A,B5: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( prod_filter @ A @ B @ A6 @ B5 ) ) @ A6 ) ).
% filtermap_fst_prod_filter
thf(fact_6885_filtermap__snd__prod__filter,axiom,
! [B: $tType,A: $tType,A6: filter @ B,B5: filter @ A] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( prod_filter @ B @ A @ A6 @ B5 ) ) @ B5 ) ).
% filtermap_snd_prod_filter
thf(fact_6886_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 ) ) )
= ( ? [N6: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ N6 @ M5 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( P @ ( product_Pair @ nat @ nat @ N3 @ M5 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_6887_filtermap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,F6: C > ( filter @ B ),B5: set @ C] :
( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F6 @ B5 ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B8: C] : ( filtermap @ B @ A @ F3 @ ( F6 @ B8 ) )
@ B5 ) ) ) ).
% filtermap_INF
thf(fact_6888_at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A] :
( ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
= ( filtermap @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ X4 @ A3 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_to_0
thf(fact_6889_le__prod__filterI,axiom,
! [A: $tType,B: $tType,F6: filter @ ( product_prod @ A @ B ),A6: filter @ A,B5: filter @ B] :
( ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ F6 ) @ A6 )
=> ( ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ F6 ) @ B5 )
=> ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ F6 @ ( prod_filter @ A @ B @ A6 @ B5 ) ) ) ) ).
% le_prod_filterI
thf(fact_6890_filterlim__INF__INF,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,J5: set @ A,I5: set @ B,F3: D > C,F6: B > ( filter @ D ),G7: A > ( filter @ C )] :
( ! [M: A] :
( ( member @ A @ M @ J5 )
=> ? [X: B] :
( ( member @ B @ X @ I5 )
& ( ord_less_eq @ ( filter @ C ) @ ( filtermap @ D @ C @ F3 @ ( F6 @ X ) ) @ ( G7 @ M ) ) ) )
=> ( filterlim @ D @ C @ F3 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ G7 @ J5 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ F6 @ I5 ) ) ) ) ).
% filterlim_INF_INF
thf(fact_6891_prod__filter__principal__singleton,axiom,
! [A: $tType,B: $tType,X3: A,F6: filter @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ F6 )
= ( filtermap @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ F6 ) ) ).
% prod_filter_principal_singleton
thf(fact_6892_prod__filter__principal__singleton2,axiom,
! [B: $tType,A: $tType,F6: filter @ A,X3: B] :
( ( prod_filter @ A @ B @ F6 @ ( principal @ B @ ( insert2 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( filtermap @ A @ ( product_prod @ A @ B )
@ ^ [A8: A] : ( product_Pair @ A @ B @ A8 @ X3 )
@ F6 ) ) ).
% prod_filter_principal_singleton2
thf(fact_6893_finite__enum__subset,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X6: set @ A,Y8: set @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( finite_card @ A @ X6 ) )
=> ( ( infini527867602293511546merate @ A @ X6 @ I3 )
= ( infini527867602293511546merate @ A @ Y8 @ I3 ) ) )
=> ( ( finite_finite2 @ A @ X6 )
=> ( ( finite_finite2 @ A @ Y8 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ X6 ) @ ( finite_card @ A @ Y8 ) )
=> ( ord_less_eq @ ( set @ A ) @ X6 @ Y8 ) ) ) ) ) ) ).
% finite_enum_subset
thf(fact_6894_uniformly__continuous__on__uniformity,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo7287701948861334536_space @ A )
& ( topolo7287701948861334536_space @ B ) )
=> ( ( topolo6026614971017936543ous_on @ A @ B )
= ( ^ [S7: set @ A,F4: A > B] :
( filterlim @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B )
@ ( product_case_prod @ A @ A @ ( product_prod @ B @ B )
@ ^ [X4: A,Y3: A] : ( product_Pair @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y3 ) ) )
@ ( topolo7806501430040627800ormity @ B )
@ ( inf_inf @ ( filter @ ( product_prod @ A @ A ) ) @ ( topolo7806501430040627800ormity @ A )
@ ( principal @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ S7
@ ^ [Uu3: A] : S7 ) ) ) ) ) ) ) ).
% uniformly_continuous_on_uniformity
thf(fact_6895_SigmaI,axiom,
! [B: $tType,A: $tType,A3: A,A6: set @ A,B2: B,B5: A > ( set @ B )] :
( ( member @ A @ A3 @ A6 )
=> ( ( member @ B @ B2 @ ( B5 @ A3 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A6 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_6896_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A6: set @ A,B5: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A6 @ B5 ) )
= ( ( member @ A @ A3 @ A6 )
& ( member @ B @ B2 @ ( B5 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_6897_Field__square,axiom,
! [A: $tType,X3: set @ A] :
( ( field2 @ A
@ ( product_Sigma @ A @ A @ X3
@ ^ [Uu3: A] : X3 ) )
= X3 ) ).
% Field_square
thf(fact_6898_Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A8: A,B8: B] :
( ( P @ A8 )
& ( Q @ B8 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [Uu3: A] : ( collect @ B @ Q ) ) ) ).
% Collect_case_prod
thf(fact_6899_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_6900_Compl__Times__UNIV1,axiom,
! [B: $tType,A: $tType,A6: set @ B] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : A6 ) )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : ( uminus_uminus @ ( set @ B ) @ A6 ) ) ) ).
% Compl_Times_UNIV1
thf(fact_6901_Compl__Times__UNIV2,axiom,
! [B: $tType,A: $tType,A6: set @ A] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) )
= ( product_Sigma @ A @ B @ ( uminus_uminus @ ( set @ A ) @ A6 )
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) ) ).
% Compl_Times_UNIV2
thf(fact_6902_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A6: set @ A] :
( ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_6903_Times__empty,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: set @ B] :
( ( ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ( B5
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_6904_UNIV__Times__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% UNIV_Times_UNIV
thf(fact_6905_fst__image__times,axiom,
! [B: $tType,A: $tType,B5: set @ B,A6: set @ A] :
( ( ( B5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 ) )
= A6 ) ) ) ).
% fst_image_times
thf(fact_6906_snd__image__times,axiom,
! [B: $tType,A: $tType,A6: set @ B,B5: set @ A] :
( ( ( A6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu3: B] : B5 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu3: B] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_6907_set__product,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs2 @ Ys ) )
= ( product_Sigma @ A @ B @ ( set2 @ A @ Xs2 )
@ ^ [Uu3: A] : ( set2 @ B @ Ys ) ) ) ).
% set_product
thf(fact_6908_insert__Times__insert,axiom,
! [B: $tType,A: $tType,A3: A,A6: set @ A,B2: B,B5: set @ B] :
( ( product_Sigma @ A @ B @ ( insert2 @ A @ A3 @ A6 )
@ ^ [Uu3: A] : ( insert2 @ B @ B2 @ B5 ) )
= ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 )
@ ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : ( insert2 @ B @ B2 @ B5 ) )
@ ( product_Sigma @ A @ B @ ( insert2 @ A @ A3 @ A6 )
@ ^ [Uu3: A] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_6909_inj__on__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F3: A > C,A6: set @ A] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F3 )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) )
= ( inj_on @ A @ C @ F3 @ A6 ) ) ).
% inj_on_apfst
thf(fact_6910_inj__on__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F3: B > C,A6: set @ B] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : A6 ) )
= ( inj_on @ B @ C @ F3 @ A6 ) ) ).
% inj_on_apsnd
thf(fact_6911_Id__on__subset__Times,axiom,
! [A: $tType,A6: set @ A] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( id_on @ A @ A6 )
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu3: A] : A6 ) ) ).
% Id_on_subset_Times
thf(fact_6912_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y3: B] :
( ( P @ X4 )
& ( Q @ X4 @ Y3 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [X4: A] : ( collect @ B @ ( Q @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_6913_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X3: A,C4: set @ A,A6: set @ B,B5: set @ B] :
( ( member @ A @ X3 @ C4 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu3: B] : C4 )
@ ( product_Sigma @ B @ A @ B5
@ ^ [Uu3: B] : C4 ) )
= ( ord_less_eq @ ( set @ B ) @ A6 @ B5 ) ) ) ).
% Times_subset_cancel2
thf(fact_6914_Sigma__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,C4: set @ A,B5: A > ( set @ B ),D4: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ ( B5 @ X5 ) @ ( D4 @ X5 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A6 @ B5 ) @ ( product_Sigma @ A @ B @ C4 @ D4 ) ) ) ) ).
% Sigma_mono
thf(fact_6915_Restr__subset,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B5
@ ^ [Uu3: A] : B5 ) )
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu3: A] : A6 ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu3: A] : A6 ) ) ) ) ).
% Restr_subset
thf(fact_6916_relcomp__subset__Sigma,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),A6: set @ A,B5: set @ B,S: set @ ( product_prod @ B @ C ),C4: set @ C] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S
@ ( product_Sigma @ B @ C @ B5
@ ^ [Uu3: B] : C4 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R2 @ S )
@ ( product_Sigma @ A @ C @ A6
@ ^ [Uu3: A] : C4 ) ) ) ) ).
% relcomp_subset_Sigma
thf(fact_6917_le__enumerate,axiom,
! [S3: set @ nat,N: nat] :
( ~ ( finite_finite2 @ nat @ S3 )
=> ( ord_less_eq @ nat @ N @ ( infini527867602293511546merate @ nat @ S3 @ N ) ) ) ).
% le_enumerate
thf(fact_6918_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_6919_times__eq__iff,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: set @ B,C4: set @ A,D4: set @ B] :
( ( ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 )
= ( product_Sigma @ A @ B @ C4
@ ^ [Uu3: A] : D4 ) )
= ( ( ( A6 = C4 )
& ( B5 = D4 ) )
| ( ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ( B5
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C4
= ( bot_bot @ ( set @ A ) ) )
| ( D4
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_6920_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I5: set @ A,X6: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I5 @ X6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ I5 )
=> ( ( X6 @ X4 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_6921_SigmaE,axiom,
! [A: $tType,B: $tType,C3: product_prod @ A @ B,A6: set @ A,B5: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A6 @ B5 ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ! [Y4: B] :
( ( member @ B @ Y4 @ ( B5 @ X5 ) )
=> ( C3
!= ( product_Pair @ A @ B @ X5 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_6922_SigmaD1,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A6: set @ A,B5: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A6 @ B5 ) )
=> ( member @ A @ A3 @ A6 ) ) ).
% SigmaD1
thf(fact_6923_SigmaD2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A6: set @ A,B5: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A6 @ B5 ) )
=> ( member @ B @ B2 @ ( B5 @ A3 ) ) ) ).
% SigmaD2
thf(fact_6924_SigmaE2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A6: set @ A,B5: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A6 @ B5 ) )
=> ~ ( ( member @ A @ A3 @ A6 )
=> ~ ( member @ B @ B2 @ ( B5 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_6925_Sigma__Union,axiom,
! [B: $tType,A: $tType,X6: set @ ( set @ A ),B5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ X6 ) @ B5 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( set @ A ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A7: set @ A] : ( product_Sigma @ A @ B @ A7 @ B5 )
@ X6 ) ) ) ).
% Sigma_Union
thf(fact_6926_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X3: product_prod @ A @ B,A6: set @ A,B5: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X3
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 ) )
= ( ( member @ A @ ( product_fst @ A @ B @ X3 ) @ A6 )
& ( member @ B @ ( product_snd @ A @ B @ X3 ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_6927_member__product,axiom,
! [B: $tType,A: $tType,X3: product_prod @ A @ B,A6: set @ A,B5: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X3 @ ( product_product @ A @ B @ A6 @ B5 ) )
= ( member @ ( product_prod @ A @ B ) @ X3
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 ) ) ) ).
% member_product
thf(fact_6928_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A7: set @ A,B6: set @ B] :
( product_Sigma @ A @ B @ A7
@ ^ [Uu3: A] : B6 ) ) ) ).
% Product_Type.product_def
thf(fact_6929_Sigma__cong,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ A,C4: A > ( set @ B ),D4: A > ( set @ B )] :
( ( A6 = B5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B5 )
=> ( ( C4 @ X5 )
= ( D4 @ X5 ) ) )
=> ( ( product_Sigma @ A @ B @ A6 @ C4 )
= ( product_Sigma @ A @ B @ B5 @ D4 ) ) ) ) ).
% Sigma_cong
thf(fact_6930_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X3: A,C4: set @ A,A6: set @ B,B5: set @ B] :
( ( member @ A @ X3 @ C4 )
=> ( ( ( product_Sigma @ B @ A @ A6
@ ^ [Uu3: B] : C4 )
= ( product_Sigma @ B @ A @ B5
@ ^ [Uu3: B] : C4 ) )
= ( A6 = B5 ) ) ) ).
% Times_eq_cancel2
thf(fact_6931_Sigma__Un__distrib1,axiom,
! [B: $tType,A: $tType,I5: set @ A,J5: set @ A,C4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ I5 @ J5 ) @ C4 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ C4 ) @ ( product_Sigma @ A @ B @ J5 @ C4 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_6932_Times__Un__distrib1,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ A,C4: set @ B] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A6 @ B5 )
@ ^ [Uu3: A] : C4 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : C4 )
@ ( product_Sigma @ A @ B @ B5
@ ^ [Uu3: A] : C4 ) ) ) ).
% Times_Un_distrib1
thf(fact_6933_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A6: A > ( set @ B ),B5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I4: A] : ( sup_sup @ ( set @ B ) @ ( A6 @ I4 ) @ ( B5 @ I4 ) ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A6 ) @ ( product_Sigma @ A @ B @ I5 @ B5 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_6934_Sigma__Diff__distrib1,axiom,
! [B: $tType,A: $tType,I5: set @ A,J5: set @ A,C4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ I5 @ J5 ) @ C4 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ C4 ) @ ( product_Sigma @ A @ B @ J5 @ C4 ) ) ) ).
% Sigma_Diff_distrib1
thf(fact_6935_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A6: A > ( set @ B ),B5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I4: A] : ( minus_minus @ ( set @ B ) @ ( A6 @ I4 ) @ ( B5 @ I4 ) ) )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A6 ) @ ( product_Sigma @ A @ B @ I5 @ B5 ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_6936_Times__Diff__distrib1,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ A,C4: set @ B] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A6 @ B5 )
@ ^ [Uu3: A] : C4 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : C4 )
@ ( product_Sigma @ A @ B @ B5
@ ^ [Uu3: A] : C4 ) ) ) ).
% Times_Diff_distrib1
thf(fact_6937_Times__Int__distrib1,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ A,C4: set @ B] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A6 @ B5 )
@ ^ [Uu3: A] : C4 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : C4 )
@ ( product_Sigma @ A @ B @ B5
@ ^ [Uu3: A] : C4 ) ) ) ).
% Times_Int_distrib1
thf(fact_6938_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A6: A > ( set @ B ),B5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I4: A] : ( inf_inf @ ( set @ B ) @ ( A6 @ I4 ) @ ( B5 @ I4 ) ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A6 ) @ ( product_Sigma @ A @ B @ I5 @ B5 ) ) ) ).
% Sigma_Int_distrib2
thf(fact_6939_Times__Int__Times,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B,C4: set @ A,D4: set @ B] :
( ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 )
@ ( product_Sigma @ A @ B @ C4
@ ^ [Uu3: A] : D4 ) )
= ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A6 @ C4 )
@ ^ [Uu3: A] : ( inf_inf @ ( set @ B ) @ B5 @ D4 ) ) ) ).
% Times_Int_Times
thf(fact_6940_Sigma__Int__distrib1,axiom,
! [B: $tType,A: $tType,I5: set @ A,J5: set @ A,C4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ I5 @ J5 ) @ C4 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ C4 ) @ ( product_Sigma @ A @ B @ J5 @ C4 ) ) ) ).
% Sigma_Int_distrib1
thf(fact_6941_times__subset__iff,axiom,
! [A: $tType,B: $tType,A6: set @ A,C4: set @ B,B5: set @ A,D4: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : C4 )
@ ( product_Sigma @ A @ B @ B5
@ ^ [Uu3: A] : D4 ) )
= ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ( C4
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
& ( ord_less_eq @ ( set @ B ) @ C4 @ D4 ) ) ) ) ).
% times_subset_iff
thf(fact_6942_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu3: A] : A6 ) )
=> ( ( A3 = B2 )
| ( member @ A @ A3 @ A6 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_6943_Field__Restr__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ord_less_eq @ ( set @ A )
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu3: A] : A6 ) ) )
@ A6 ) ).
% Field_Restr_subset
thf(fact_6944_wfI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu3: A] : B5 ) )
=> ( ! [X5: A,P7: A > $o] :
( ! [Xa: A] :
( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa ) @ R2 )
=> ( P7 @ Y4 ) )
=> ( P7 @ Xa ) )
=> ( ( member @ A @ X5 @ A6 )
=> ( ( member @ A @ X5 @ B5 )
=> ( P7 @ X5 ) ) ) )
=> ( wf @ A @ R2 ) ) ) ).
% wfI
thf(fact_6945_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: A > ( set @ B )] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ( B5 @ X4 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ( finite_finite2 @ B @ ( B5 @ A5 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A6 @ B5 ) ) ) ) ).
% finite_SigmaI2
thf(fact_6946_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( finite_finite2 @ A @ A6 ) ) ) ).
% finite_cartesian_productD1
thf(fact_6947_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 ) )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( finite_finite2 @ B @ B5 ) ) ) ).
% finite_cartesian_productD2
thf(fact_6948_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 ) )
= ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ( B5
= ( bot_bot @ ( set @ B ) ) )
| ( ( finite_finite2 @ A @ A6 )
& ( finite_finite2 @ B @ B5 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_6949_Image__subset,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),A6: set @ A,B5: set @ B,C4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R2 @ C4 ) @ B5 ) ) ).
% Image_subset
thf(fact_6950_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: A > ( set @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A6 @ B5 ) )
= ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ( B5 @ X4 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_6951_UN__Times__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,E5: C > ( set @ A ),F6: D > ( set @ B ),A6: set @ C,B5: set @ D] :
( ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( product_prod @ C @ D ) @ ( set @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ C @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A8: C,B8: D] :
( product_Sigma @ A @ B @ ( E5 @ A8 )
@ ^ [Uu3: A] : ( F6 @ B8 ) ) )
@ ( product_Sigma @ C @ D @ A6
@ ^ [Uu3: C] : B5 ) ) )
= ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ E5 @ A6 ) )
@ ^ [Uu3: A] : ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ D @ ( set @ B ) @ F6 @ B5 ) ) ) ) ).
% UN_Times_distrib
thf(fact_6952_filtermap__image__finite__subsets__at__top,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ( filtermap @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F3 ) @ ( finite5375528669736107172at_top @ A @ A6 ) )
= ( finite5375528669736107172at_top @ B @ ( image2 @ A @ B @ F3 @ A6 ) ) ) ) ).
% filtermap_image_finite_subsets_at_top
thf(fact_6953_swap__product,axiom,
! [B: $tType,A: $tType,A6: set @ B,B5: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I4: B,J3: A] : ( product_Pair @ A @ B @ J3 @ I4 ) )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu3: B] : B5 ) )
= ( product_Sigma @ A @ B @ B5
@ ^ [Uu3: A] : A6 ) ) ).
% swap_product
thf(fact_6954_map__prod__surj__on,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F3: B > A,A6: set @ B,A11: set @ A,G3: D > C,B5: set @ D,B13: set @ C] :
( ( ( image2 @ B @ A @ F3 @ A6 )
= A11 )
=> ( ( ( image2 @ D @ C @ G3 @ B5 )
= B13 )
=> ( ( image2 @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F3 @ G3 )
@ ( product_Sigma @ B @ D @ A6
@ ^ [Uu3: B] : B5 ) )
= ( product_Sigma @ A @ C @ A11
@ ^ [Uu3: A] : B13 ) ) ) ) ).
% map_prod_surj_on
thf(fact_6955_enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S3 )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) ) ) ) ) ).
% enumerate_step
thf(fact_6956_map__prod__inj__on,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,F3: A > B,A6: set @ A,G3: C > D,B5: set @ C] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ( inj_on @ C @ D @ G3 @ B5 )
=> ( inj_on @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F3 @ G3 )
@ ( product_Sigma @ A @ C @ A6
@ ^ [Uu3: A] : B5 ) ) ) ) ).
% map_prod_inj_on
thf(fact_6957_principal__prod__principal,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ A6 ) @ ( principal @ B @ B5 ) )
= ( principal @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 ) ) ) ).
% principal_prod_principal
thf(fact_6958_Sigma__Image,axiom,
! [A: $tType,B: $tType,A6: set @ B,B5: B > ( set @ A ),X6: set @ B] :
( ( image @ B @ A @ ( product_Sigma @ B @ A @ A6 @ B5 ) @ X6 )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ ( inf_inf @ ( set @ B ) @ X6 @ A6 ) ) ) ) ).
% Sigma_Image
thf(fact_6959_snd__image__Sigma,axiom,
! [A: $tType,B: $tType,A6: set @ B,B5: B > ( set @ A )] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( product_Sigma @ B @ A @ A6 @ B5 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% snd_image_Sigma
thf(fact_6960_subset__fst__imageI,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B,S3: set @ ( product_prod @ A @ B ),Y: B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 )
@ S3 )
=> ( ( member @ B @ Y @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S3 ) ) ) ) ).
% subset_fst_imageI
thf(fact_6961_subset__snd__imageI,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B,S3: set @ ( product_prod @ A @ B ),X3: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 )
@ S3 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ B5 @ ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S3 ) ) ) ) ).
% subset_snd_imageI
thf(fact_6962_subset__fst__snd,axiom,
! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6
@ ( product_Sigma @ A @ B @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 )
@ ^ [Uu3: A] : ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) ) ) ).
% subset_fst_snd
thf(fact_6963_card__cartesian__product__singleton,axiom,
! [A: $tType,B: $tType,X3: A,A6: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu3: A] : A6 ) )
= ( finite_card @ B @ A6 ) ) ).
% card_cartesian_product_singleton
thf(fact_6964_image__paired__Times,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F3: C > A,G3: D > B,A6: set @ C,B5: set @ D] :
( ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X4: C,Y3: D] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ Y3 ) ) )
@ ( product_Sigma @ C @ D @ A6
@ ^ [Uu3: C] : B5 ) )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F3 @ A6 )
@ ^ [Uu3: A] : ( image2 @ D @ B @ G3 @ B5 ) ) ) ).
% image_paired_Times
thf(fact_6965_lists__length__Suc__eq,axiom,
! [A: $tType,A6: set @ A,N: nat] :
( ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A6 )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) ) ) )
= ( image2 @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs: list @ A,N3: A] : ( cons @ A @ N3 @ Xs ) )
@ ( product_Sigma @ ( list @ A ) @ A
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A6 )
& ( ( size_size @ ( list @ A ) @ Xs )
= N ) ) )
@ ^ [Uu3: list @ A] : A6 ) ) ) ).
% lists_length_Suc_eq
thf(fact_6966_finite__le__enumerate,axiom,
! [S3: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ S3 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ nat @ S3 ) )
=> ( ord_less_eq @ nat @ N @ ( infini527867602293511546merate @ nat @ S3 @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_6967_pairs__le__eq__Sigma,axiom,
! [M2: nat] :
( ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ M2 ) ) )
= ( product_Sigma @ nat @ nat @ ( set_ord_atMost @ nat @ M2 )
@ ^ [R5: nat] : ( set_ord_atMost @ nat @ ( minus_minus @ nat @ M2 @ R5 ) ) ) ) ).
% pairs_le_eq_Sigma
thf(fact_6968_Sigma__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B )
= ( ^ [A7: set @ A,B6: A > ( set @ B )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y3: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
@ ( B6 @ X4 ) ) )
@ A7 ) ) ) ) ).
% Sigma_def
thf(fact_6969_product__fold,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: A,Z4: set @ ( product_prod @ A @ B )] :
( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y3: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) )
@ Z4
@ B5 )
@ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
@ A6 ) ) ) ) ).
% product_fold
thf(fact_6970_finite__enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ( finite_finite2 @ A @ S3 )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S3 ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_6971_enumerate__Suc_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) )
= ( infini527867602293511546merate @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert2 @ A @ ( infini527867602293511546merate @ A @ S3 @ ( zero_zero @ nat ) ) @ ( bot_bot @ ( set @ A ) ) ) ) @ N ) ) ) ).
% enumerate_Suc'
thf(fact_6972_finite__enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ( finite_finite2 @ A @ S3 )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S3 ) )
=> ( ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S7: A] :
( ( member @ A @ S7 @ S3 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ S7 ) ) ) ) ) ) ) ).
% finite_enumerate_Suc''
thf(fact_6973_enumerate__Suc,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) )
= ( infini527867602293511546merate @ A
@ ( minus_minus @ ( set @ A ) @ S3
@ ( insert2 @ A
@ ( ord_Least @ A
@ ^ [N3: A] : ( member @ A @ N3 @ S3 ) )
@ ( bot_bot @ ( set @ A ) ) ) )
@ N ) ) ) ).
% enumerate_Suc
thf(fact_6974_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( zero_zero @ nat ) ) ) ).
% Least_eq_0
thf(fact_6975_Least__Suc2,axiom,
! [P: nat > $o,N: nat,Q: nat > $o,M2: nat] :
( ( P @ N )
=> ( ( Q @ M2 )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ! [K: nat] :
( ( P @ ( suc @ K ) )
= ( Q @ K ) )
=> ( ( ord_Least @ nat @ P )
= ( suc @ ( ord_Least @ nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_6976_not__less__Least,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [K2: A,P: A > $o] :
( ( ord_less @ A @ K2 @ ( ord_Least @ A @ P ) )
=> ~ ( P @ K2 ) ) ) ).
% not_less_Least
thf(fact_6977_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K2: A] :
( ( P @ K2 )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI
thf(fact_6978_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_ex
thf(fact_6979_LeastI__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI_ex
thf(fact_6980_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2
thf(fact_6981_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( ! [A5: A] :
( ( P @ A5 )
=> ( ! [B10: A] :
( ( P @ B10 )
=> ( ord_less_eq @ A @ A5 @ B10 ) )
=> ( Q @ A5 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_6982_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [A5: A] :
( ( P @ A5 )
=> ( ! [B10: A] :
( ( P @ B10 )
=> ( ord_less_eq @ A @ A5 @ B10 ) )
=> ( Q @ A5 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_6983_Least__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ X3 @ Y4 ) )
=> ( ( ord_Least @ A @ P )
= X3 ) ) ) ) ).
% Least_equality
thf(fact_6984_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A,Q: A > $o] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ X3 @ Y4 ) )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ A @ X5 @ Y6 ) )
=> ( Q @ X5 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_6985_Least1__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,Z2: A] :
( ? [X: A] :
( ( P @ X )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ X @ Y4 ) )
& ! [Y4: A] :
( ( ( P @ Y4 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y4 @ Ya2 ) ) )
=> ( Y4 = X ) ) )
=> ( ( P @ Z2 )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z2 ) ) ) ) ).
% Least1_le
thf(fact_6986_Least1I,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o] :
( ? [X: A] :
( ( P @ X )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ X @ Y4 ) )
& ! [Y4: A] :
( ( ( P @ Y4 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y4 @ Ya2 ) ) )
=> ( Y4 = X ) ) )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% Least1I
thf(fact_6987_Least__le,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K2: A] :
( ( P @ K2 )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ K2 ) ) ) ).
% Least_le
thf(fact_6988_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( suc
@ ( ord_Least @ nat
@ ^ [M5: nat] : ( P @ ( suc @ M5 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_6989_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,S3: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ? [X: A] :
( ( member @ A @ X @ S3 )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ S3 )
=> ( ord_less_eq @ A @ X @ Xa3 ) ) )
=> ( ( ord_Least @ B
@ ^ [Y3: B] : ( member @ B @ Y3 @ ( image2 @ A @ B @ F3 @ S3 ) ) )
= ( F3
@ ( ord_Least @ A
@ ^ [X4: A] : ( member @ A @ X4 @ S3 ) ) ) ) ) ) ) ).
% Least_mono
thf(fact_6990_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P4: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P4 @ X4 )
& ! [Y3: A] :
( ( P4 @ Y3 )
=> ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ) ) ) ) ).
% Least_def
thf(fact_6991_enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S3 )
=> ( ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S7: A] :
( ( member @ A @ S7 @ S3 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ S7 ) ) ) ) ) ) ).
% enumerate_Suc''
thf(fact_6992_Gr__incl,axiom,
! [A: $tType,B: $tType,A6: set @ A,F3: A > B,B5: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A6 @ F3 )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 ) )
= ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A6 ) @ B5 ) ) ).
% Gr_incl
thf(fact_6993_image__split__eq__Sigma,axiom,
! [C: $tType,B: $tType,A: $tType,F3: C > A,G3: C > B,A6: set @ C] :
( ( image2 @ C @ ( product_prod @ A @ B )
@ ^ [X4: C] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ A6 )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F3 @ A6 )
@ ^ [X4: A] : ( image2 @ C @ B @ G3 @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F3 @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A6 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_6994_vimageI,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: B,B2: A,B5: set @ A] :
( ( ( F3 @ A3 )
= B2 )
=> ( ( member @ A @ B2 @ B5 )
=> ( member @ B @ A3 @ ( vimage @ B @ A @ F3 @ B5 ) ) ) ) ).
% vimageI
thf(fact_6995_vimage__eq,axiom,
! [A: $tType,B: $tType,A3: A,F3: A > B,B5: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F3 @ B5 ) )
= ( member @ B @ ( F3 @ A3 ) @ B5 ) ) ).
% vimage_eq
thf(fact_6996_vimage__ident,axiom,
! [A: $tType,Y8: set @ A] :
( ( vimage @ A @ A
@ ^ [X4: A] : X4
@ Y8 )
= Y8 ) ).
% vimage_ident
thf(fact_6997_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_6998_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_6999_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_7000_vimage__Int,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ B,B5: set @ B] :
( ( vimage @ A @ B @ F3 @ ( inf_inf @ ( set @ B ) @ A6 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A6 ) @ ( vimage @ A @ B @ F3 @ B5 ) ) ) ).
% vimage_Int
thf(fact_7001_vimage__Un,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ B,B5: set @ B] :
( ( vimage @ A @ B @ F3 @ ( sup_sup @ ( set @ B ) @ A6 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A6 ) @ ( vimage @ A @ B @ F3 @ B5 ) ) ) ).
% vimage_Un
thf(fact_7002_filtercomap__principal,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ B] :
( ( filtercomap @ A @ B @ F3 @ ( principal @ B @ A6 ) )
= ( principal @ A @ ( vimage @ A @ B @ F3 @ A6 ) ) ) ).
% filtercomap_principal
thf(fact_7003_vimage__const,axiom,
! [B: $tType,A: $tType,C3: B,A6: set @ B] :
( ( ( member @ B @ C3 @ A6 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : C3
@ A6 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ C3 @ A6 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : C3
@ A6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% vimage_const
thf(fact_7004_image__vimage__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ A] :
( ( image2 @ B @ A @ F3 @ ( vimage @ B @ A @ F3 @ A6 ) )
= ( inf_inf @ ( set @ A ) @ A6 @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% image_vimage_eq
thf(fact_7005_vimage__if,axiom,
! [B: $tType,A: $tType,C3: B,A6: set @ B,D3: B,B5: set @ A] :
( ( ( member @ B @ C3 @ A6 )
=> ( ( ( member @ B @ D3 @ A6 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ B5 ) @ C3 @ D3 )
@ A6 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ D3 @ A6 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ B5 ) @ C3 @ D3 )
@ A6 )
= B5 ) ) ) )
& ( ~ ( member @ B @ C3 @ A6 )
=> ( ( ( member @ B @ D3 @ A6 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ B5 ) @ C3 @ D3 )
@ A6 )
= ( uminus_uminus @ ( set @ A ) @ B5 ) ) )
& ( ~ ( member @ B @ D3 @ A6 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ B5 ) @ C3 @ D3 )
@ A6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% vimage_if
thf(fact_7006_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X3: B,A6: set @ B,F3: B > ( set @ A )] :
( ( ( member @ B @ X3 @ A6 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 ) @ ( product_Sigma @ B @ A @ A6 @ F3 ) )
= ( F3 @ X3 ) ) )
& ( ~ ( member @ B @ X3 @ A6 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 ) @ ( product_Sigma @ B @ A @ A6 @ F3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pair_vimage_Sigma
thf(fact_7007_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > ( product_prod @ B @ C ),A6: set @ B,B5: set @ C] :
( ( vimage @ A @ ( product_prod @ B @ C ) @ F3
@ ( product_Sigma @ B @ C @ A6
@ ^ [Uu3: B] : B5 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F3 ) @ A6 ) @ ( vimage @ A @ C @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F3 ) @ B5 ) ) ) ).
% vimage_Times
thf(fact_7008_vimage__Compl,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ B] :
( ( vimage @ A @ B @ F3 @ ( uminus_uminus @ ( set @ B ) @ A6 ) )
= ( uminus_uminus @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A6 ) ) ) ).
% vimage_Compl
thf(fact_7009_vimage__subsetD,axiom,
! [A: $tType,B: $tType,F3: B > A,B5: set @ A,A6: set @ B] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ B5 ) @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ).
% vimage_subsetD
thf(fact_7010_vimage__image__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A] :
( ( vimage @ A @ B @ F3 @ ( image2 @ A @ B @ F3 @ A6 ) )
= ( collect @ A
@ ^ [Y3: A] :
? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ( F3 @ X4 )
= ( F3 @ Y3 ) ) ) ) ) ).
% vimage_image_eq
thf(fact_7011_image__vimage__subset,axiom,
! [B: $tType,A: $tType,F3: B > A,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ ( vimage @ B @ A @ F3 @ A6 ) ) @ A6 ) ).
% image_vimage_subset
thf(fact_7012_image__subset__iff__subset__vimage,axiom,
! [B: $tType,A: $tType,F3: B > A,A6: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A6 ) @ B5 )
= ( ord_less_eq @ ( set @ B ) @ A6 @ ( vimage @ B @ A @ F3 @ B5 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_7013_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F3: B > A,A6: set @ A] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( vimage @ B @ A @ F3 @ A6 )
= ( bot_bot @ ( set @ B ) ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% surj_vimage_empty
thf(fact_7014_vimage__Suc__insert__0,axiom,
! [A6: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert2 @ nat @ ( zero_zero @ nat ) @ A6 ) )
= ( vimage @ nat @ nat @ suc @ A6 ) ) ).
% vimage_Suc_insert_0
thf(fact_7015_vimage__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ A,F3: B > A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ A6 ) @ ( vimage @ B @ A @ F3 @ B5 ) ) ) ).
% vimage_mono
thf(fact_7016_subset__vimage__iff,axiom,
! [B: $tType,A: $tType,A6: set @ A,F3: A > B,B5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( vimage @ A @ B @ F3 @ B5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( member @ B @ ( F3 @ X4 ) @ B5 ) ) ) ) ).
% subset_vimage_iff
thf(fact_7017_vimage__insert,axiom,
! [A: $tType,B: $tType,F3: A > B,A3: B,B5: set @ B] :
( ( vimage @ A @ B @ F3 @ ( insert2 @ B @ A3 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( vimage @ A @ B @ F3 @ B5 ) ) ) ).
% vimage_insert
thf(fact_7018_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A3: A,F3: A > B,B2: B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F3 @ ( insert2 @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( ( F3 @ A3 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_7019_GrD1,axiom,
! [B: $tType,A: $tType,X3: A,Fx: B,A6: set @ A,F3: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Fx ) @ ( bNF_Gr @ A @ B @ A6 @ F3 ) )
=> ( member @ A @ X3 @ A6 ) ) ).
% GrD1
thf(fact_7020_GrD2,axiom,
! [A: $tType,B: $tType,X3: A,Fx: B,A6: set @ A,F3: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Fx ) @ ( bNF_Gr @ A @ B @ A6 @ F3 ) )
=> ( ( F3 @ X3 )
= Fx ) ) ).
% GrD2
thf(fact_7021_finite__vimage__Suc__iff,axiom,
! [F6: set @ nat] :
( ( finite_finite2 @ nat @ ( vimage @ nat @ nat @ suc @ F6 ) )
= ( finite_finite2 @ nat @ F6 ) ) ).
% finite_vimage_Suc_iff
thf(fact_7022_vimageD,axiom,
! [A: $tType,B: $tType,A3: A,F3: A > B,A6: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F3 @ A6 ) )
=> ( member @ B @ ( F3 @ A3 ) @ A6 ) ) ).
% vimageD
thf(fact_7023_vimageE,axiom,
! [A: $tType,B: $tType,A3: A,F3: A > B,B5: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F3 @ B5 ) )
=> ( member @ B @ ( F3 @ A3 ) @ B5 ) ) ).
% vimageE
thf(fact_7024_vimageI2,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: B,A6: set @ A] :
( ( member @ A @ ( F3 @ A3 ) @ A6 )
=> ( member @ B @ A3 @ ( vimage @ B @ A @ F3 @ A6 ) ) ) ).
% vimageI2
thf(fact_7025_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: B > $o,F3: A > B,Q: A > $o] :
( ! [X5: A] :
( ( P @ ( F3 @ X5 ) )
= ( Q @ X5 ) )
=> ( ( vimage @ A @ B @ F3 @ ( collect @ B @ P ) )
= ( collect @ A @ Q ) ) ) ).
% vimage_Collect
thf(fact_7026_vimage__def,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F4: A > B,B6: set @ B] :
( collect @ A
@ ^ [X4: A] : ( member @ B @ ( F4 @ X4 ) @ B6 ) ) ) ) ).
% vimage_def
thf(fact_7027_vimage__Diff,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ B,B5: set @ B] :
( ( vimage @ A @ B @ F3 @ ( minus_minus @ ( set @ B ) @ A6 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A6 ) @ ( vimage @ A @ B @ F3 @ B5 ) ) ) ).
% vimage_Diff
thf(fact_7028_vimage__Suc__insert__Suc,axiom,
! [N: nat,A6: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert2 @ nat @ ( suc @ N ) @ A6 ) )
= ( insert2 @ nat @ N @ ( vimage @ nat @ nat @ suc @ A6 ) ) ) ).
% vimage_Suc_insert_Suc
thf(fact_7029_vimage__inter__cong,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B,G3: A > B,Y: set @ B] :
( ! [W2: A] :
( ( member @ A @ W2 @ S3 )
=> ( ( F3 @ W2 )
= ( G3 @ W2 ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ Y ) @ S3 )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G3 @ Y ) @ S3 ) ) ) ).
% vimage_inter_cong
thf(fact_7030_vimage__fst,axiom,
! [B: $tType,A: $tType,A6: set @ A] :
( ( vimage @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 )
= ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) ) ).
% vimage_fst
thf(fact_7031_vimage__snd,axiom,
! [B: $tType,A: $tType,A6: set @ B] :
( ( vimage @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : A6 ) ) ).
% vimage_snd
thf(fact_7032_finite__vimageD_H,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ B] :
( ( finite_finite2 @ A @ ( vimage @ A @ B @ F3 @ A6 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ ( image2 @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) )
=> ( finite_finite2 @ B @ A6 ) ) ) ).
% finite_vimageD'
thf(fact_7033_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F3: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ~ ( finite_finite2 @ B @ A6 )
=> ~ ! [Y4: A] :
( ( member @ A @ Y4 @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( finite_finite2 @ B @ ( vimage @ B @ A @ F3 @ ( insert2 @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_7034_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F3: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ~ ( finite_finite2 @ B @ A6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( image2 @ B @ A @ F3 @ A6 ) )
& ~ ( finite_finite2 @ B @ ( vimage @ B @ A @ F3 @ ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_7035_vimage__subsetI,axiom,
! [B: $tType,A: $tType,F3: A > B,B5: set @ B,A6: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( image2 @ A @ B @ F3 @ A6 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ B5 ) @ A6 ) ) ) ).
% vimage_subsetI
thf(fact_7036_finite__finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F6: set @ A,H: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ F6 )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ F6 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H @ ( insert2 @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A6 ) ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H @ F6 ) @ A6 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_7037_countable__vimage,axiom,
! [B: $tType,A: $tType,B5: set @ A,F3: B > A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ( countable_countable @ B @ ( vimage @ B @ A @ F3 @ B5 ) )
=> ( countable_countable @ A @ B5 ) ) ) ).
% countable_vimage
thf(fact_7038_vimage__subset__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,B5: set @ B,A6: set @ A] :
( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ B5 ) @ A6 )
= ( ord_less_eq @ ( set @ B ) @ B5 @ ( image2 @ A @ B @ F3 @ A6 ) ) ) ) ).
% vimage_subset_eq
thf(fact_7039_vimage__eq__UN,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F4: A > B,B6: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : ( vimage @ A @ B @ F4 @ ( insert2 @ B @ Y3 @ ( bot_bot @ ( set @ B ) ) ) )
@ B6 ) ) ) ) ).
% vimage_eq_UN
thf(fact_7040_Gr__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr @ A @ B )
= ( ^ [A7: set @ A,F4: A > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A8: A] :
( ( Uu3
= ( product_Pair @ A @ B @ A8 @ ( F4 @ A8 ) ) )
& ( member @ A @ A8 @ A7 ) ) ) ) ) ).
% Gr_def
thf(fact_7041_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ~ ( finite_finite2 @ B @ A6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( image2 @ B @ A @ F3 @ A6 ) )
& ~ ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) @ A6 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_7042_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ~ ( finite_finite2 @ B @ A6 )
=> ~ ! [Y4: A] :
( ( member @ A @ Y4 @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ ( insert2 @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A6 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_7043_card__vimage__inj,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ ( image2 @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( finite_card @ A @ ( vimage @ A @ B @ F3 @ A6 ) )
= ( finite_card @ B @ A6 ) ) ) ) ).
% card_vimage_inj
thf(fact_7044_card__vimage__inj__on__le,axiom,
! [A: $tType,B: $tType,F3: A > B,D4: set @ A,A6: set @ B] :
( ( inj_on @ A @ B @ F3 @ D4 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A6 ) @ D4 ) ) @ ( finite_card @ B @ A6 ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_7045_set__decode__div__2,axiom,
! [X3: nat] :
( ( nat_set_decode @ ( divide_divide @ nat @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( vimage @ nat @ nat @ suc @ ( nat_set_decode @ X3 ) ) ) ).
% set_decode_div_2
thf(fact_7046_set__encode__vimage__Suc,axiom,
! [A6: set @ nat] :
( ( nat_set_encode @ ( vimage @ nat @ nat @ suc @ A6 ) )
= ( divide_divide @ nat @ ( nat_set_encode @ A6 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% set_encode_vimage_Suc
thf(fact_7047_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
@ ^ [X4: A] :
( ( F3 @ X4 )
= A3 ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_vimage_singleton
thf(fact_7048_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,A3: B] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) @ A6 )
@ ( insert2 @ A
@ ( the @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ( F3 @ X4 )
= A3 ) ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_on_vimage_singleton
thf(fact_7049_Restr__natLeq,axiom,
! [N: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat
@ ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N ) )
@ ^ [Uu3: nat] :
( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N ) ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y3: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X4 @ Y3 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_7050_relation__of__def,axiom,
! [A: $tType] :
( ( order_relation_of @ A )
= ( ^ [P4: A > A > $o,A7: set @ A] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A8: A,B8: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B8 )
@ ( product_Sigma @ A @ A @ A7
@ ^ [Uu3: A] : A7 ) )
& ( P4 @ A8 @ B8 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_7051_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less_eq @ nat ) ) ) ) ).
% natLeq_def
thf(fact_7052_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
@ ^ [X4: nat,Y3: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X4 @ Y3 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_7053_dropWhile__neq__rev,axiom,
! [A: $tType,Xs2: list @ A,X3: A] :
( ( distinct @ A @ Xs2 )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( dropWhile @ A
@ ^ [Y3: A] : Y3 != X3
@ ( rev @ A @ Xs2 ) )
= ( cons @ A @ X3
@ ( rev @ A
@ ( takeWhile @ A
@ ^ [Y3: A] : Y3 != X3
@ Xs2 ) ) ) ) ) ) ).
% dropWhile_neq_rev
thf(fact_7054_dropWhile__eq__Nil__conv,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( ( dropWhile @ A @ P @ Xs2 )
= ( nil @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X4 ) ) ) ) ).
% dropWhile_eq_Nil_conv
thf(fact_7055_dropWhile__append2,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o,Ys: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X5 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs2 @ Ys ) )
= ( dropWhile @ A @ P @ Ys ) ) ) ).
% dropWhile_append2
thf(fact_7056_dropWhile__append1,axiom,
! [A: $tType,X3: A,Xs2: list @ A,P: A > $o,Ys: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ~ ( P @ X3 )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs2 @ Ys ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs2 ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_7057_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_7058_sorted__dropWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( dropWhile @ A @ P @ Xs2 ) ) ) ) ).
% sorted_dropWhile
thf(fact_7059_length__dropWhile__le,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_dropWhile_le
thf(fact_7060_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_7061_underS__def,axiom,
! [A: $tType] :
( ( order_underS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A8: A] :
( collect @ A
@ ^ [B8: A] :
( ( B8 != A8 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ A8 ) @ R5 ) ) ) ) ) ).
% underS_def
thf(fact_7062_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_7063_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_7064_set__dropWhileD,axiom,
! [A: $tType,X3: A,P: A > $o,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( dropWhile @ A @ P @ Xs2 ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) ) ) ).
% set_dropWhileD
thf(fact_7065_dropWhile__cong,axiom,
! [A: $tType,L: list @ A,K2: list @ A,P: A > $o,Q: A > $o] :
( ( L = K2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ L ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( dropWhile @ A @ P @ L )
= ( dropWhile @ A @ Q @ K2 ) ) ) ) ).
% dropWhile_cong
thf(fact_7066_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_7067_takeWhile__eq__filter,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ ( dropWhile @ A @ P @ Xs2 ) ) )
=> ~ ( P @ X5 ) )
=> ( ( takeWhile @ A @ P @ Xs2 )
= ( filter2 @ A @ P @ Xs2 ) ) ) ).
% takeWhile_eq_filter
thf(fact_7068_dropWhile__eq__drop,axiom,
! [A: $tType] :
( ( dropWhile @ A )
= ( ^ [P4: A > $o,Xs: list @ A] : ( drop @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P4 @ Xs ) ) @ Xs ) ) ) ).
% dropWhile_eq_drop
thf(fact_7069_dropWhile__append,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o,Ys: list @ A] :
( ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X5 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs2 @ Ys ) )
= ( dropWhile @ A @ P @ Ys ) ) )
& ( ~ ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( P @ X ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs2 @ Ys ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs2 ) @ Ys ) ) ) ) ).
% dropWhile_append
thf(fact_7070_underS__incl__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A3 ) @ ( order_underS @ A @ R2 @ B2 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_7071_dropWhile__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs2: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs2 ) ) )
=> ( ( nth @ A @ ( dropWhile @ A @ P @ Xs2 ) @ J )
= ( nth @ A @ Xs2 @ ( plus_plus @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs2 ) ) ) ) ) ) ).
% dropWhile_nth
thf(fact_7072_extract__def,axiom,
! [A: $tType] :
( ( extract @ A )
= ( ^ [P4: A > $o,Xs: list @ A] :
( case_list @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ A @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y3: A,Ys3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( takeWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs ) @ ( product_Pair @ A @ ( list @ A ) @ Y3 @ Ys3 ) ) )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs ) ) ) ) ).
% extract_def
thf(fact_7073_find__dropWhile,axiom,
! [A: $tType] :
( ( find @ A )
= ( ^ [P4: A > $o,Xs: list @ A] :
( case_list @ ( option @ A ) @ A @ ( none @ A )
@ ^ [X4: A,Xa4: list @ A] : ( some @ A @ X4 )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs ) ) ) ) ).
% find_dropWhile
thf(fact_7074_partition__filter__conv,axiom,
! [A: $tType] :
( ( partition @ A )
= ( ^ [F4: A > $o,Xs: list @ A] : ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ F4 @ Xs ) @ ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ F4 ) @ Xs ) ) ) ) ).
% partition_filter_conv
thf(fact_7075_takeWhile__neq__rev,axiom,
! [A: $tType,Xs2: list @ A,X3: A] :
( ( distinct @ A @ Xs2 )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( takeWhile @ A
@ ^ [Y3: A] : Y3 != X3
@ ( rev @ A @ Xs2 ) )
= ( rev @ A
@ ( tl @ A
@ ( dropWhile @ A
@ ^ [Y3: A] : Y3 != X3
@ Xs2 ) ) ) ) ) ) ).
% takeWhile_neq_rev
thf(fact_7076_tl__upt,axiom,
! [M2: nat,N: nat] :
( ( tl @ nat @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ N ) ) ).
% tl_upt
thf(fact_7077_length__tl,axiom,
! [A: $tType,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( tl @ A @ Xs2 ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) ) ).
% length_tl
thf(fact_7078_sorted__tl,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( tl @ A @ Xs2 ) ) ) ) ).
% sorted_tl
thf(fact_7079_take__tl,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( take @ A @ N @ ( tl @ A @ Xs2 ) )
= ( tl @ A @ ( take @ A @ ( suc @ N ) @ Xs2 ) ) ) ).
% take_tl
thf(fact_7080_drop__Suc,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( drop @ A @ ( suc @ N ) @ Xs2 )
= ( drop @ A @ N @ ( tl @ A @ Xs2 ) ) ) ).
% drop_Suc
thf(fact_7081_list_Oset__sel_I2_J,axiom,
! [A: $tType,A3: list @ A,X3: A] :
( ( A3
!= ( nil @ A ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ ( tl @ A @ A3 ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ A3 ) ) ) ) ).
% list.set_sel(2)
thf(fact_7082_partition_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( partition @ A @ P @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) ) ).
% partition.simps(1)
thf(fact_7083_partition__P,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A,Yes: list @ A,No4: list @ A] :
( ( ( partition @ A @ P @ Xs2 )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No4 ) )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Yes ) )
=> ( P @ X ) )
& ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ No4 ) )
=> ~ ( P @ X ) ) ) ) ).
% partition_P
thf(fact_7084_nth__tl,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs2 ) ) )
=> ( ( nth @ A @ ( tl @ A @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_7085_partition_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X3: A,Xs2: list @ A] :
( ( partition @ A @ P @ ( cons @ A @ X3 @ Xs2 ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ^ [Yes2: list @ A,No3: list @ A] : ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Yes2 ) @ No3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ ( cons @ A @ X3 @ No3 ) ) )
@ ( partition @ A @ P @ Xs2 ) ) ) ).
% partition.simps(2)
thf(fact_7086_partition__set,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A,Yes: list @ A,No4: list @ A] :
( ( ( partition @ A @ P @ Xs2 )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No4 ) )
=> ( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Yes ) @ ( set2 @ A @ No4 ) )
= ( set2 @ A @ Xs2 ) ) ) ).
% partition_set
thf(fact_7087_take__Suc,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( take @ A @ ( suc @ N ) @ Xs2 )
= ( cons @ A @ ( hd @ A @ Xs2 ) @ ( take @ A @ N @ ( tl @ A @ Xs2 ) ) ) ) ) ).
% take_Suc
thf(fact_7088_inv__image__partition,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o,Ys: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X5 ) )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ ( set2 @ A @ Ys ) )
=> ~ ( P @ Y4 ) )
=> ( ( vimage @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( partition @ A @ P ) @ ( insert2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) )
= ( shuffles @ A @ Xs2 @ Ys ) ) ) ) ).
% inv_image_partition
thf(fact_7089_Nitpick_Osize__list__simp_I1_J,axiom,
! [A: $tType] :
( ( size_list @ A )
= ( ^ [F4: A > nat,Xs: list @ A] :
( if @ nat
@ ( Xs
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( plus_plus @ nat @ ( F4 @ ( hd @ A @ Xs ) ) @ ( size_list @ A @ F4 @ ( tl @ A @ Xs ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_7090_Nitpick_Osize__list__simp_I2_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( ^ [Xs: list @ A] :
( if @ nat
@ ( Xs
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_7091_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_7092_possible__bit__def,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se6407376104438227557le_bit @ A )
= ( ^ [Tyrep: itself @ A,N3: nat] :
( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
!= ( zero_zero @ A ) ) ) ) ) ).
% possible_bit_def
thf(fact_7093_map__add__find__right,axiom,
! [B: $tType,A: $tType,N: B > ( option @ A ),K2: B,Xx: A,M2: B > ( option @ A )] :
( ( ( N @ K2 )
= ( some @ A @ Xx ) )
=> ( ( map_add @ B @ A @ M2 @ N @ K2 )
= ( some @ A @ Xx ) ) ) ).
% map_add_find_right
thf(fact_7094_map__add__eq__empty__iff,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B ),G3: A > ( option @ B )] :
( ( ( map_add @ A @ B @ F3 @ G3 )
= ( ^ [X4: A] : ( none @ B ) ) )
= ( ( F3
= ( ^ [X4: A] : ( none @ B ) ) )
& ( G3
= ( ^ [X4: A] : ( none @ B ) ) ) ) ) ).
% map_add_eq_empty_iff
thf(fact_7095_map__add__None,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),N: B > ( option @ A ),K2: B] :
( ( ( map_add @ B @ A @ M2 @ N @ K2 )
= ( none @ A ) )
= ( ( ( N @ K2 )
= ( none @ A ) )
& ( ( M2 @ K2 )
= ( none @ A ) ) ) ) ).
% map_add_None
thf(fact_7096_empty__map__add,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( map_add @ A @ B
@ ^ [X4: A] : ( none @ B )
@ M2 )
= M2 ) ).
% empty_map_add
thf(fact_7097_map__add__empty,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( map_add @ A @ B @ M2
@ ^ [X4: A] : ( none @ B ) )
= M2 ) ).
% map_add_empty
thf(fact_7098_empty__eq__map__add__iff,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B ),G3: A > ( option @ B )] :
( ( ( ^ [X4: A] : ( none @ B ) )
= ( map_add @ A @ B @ F3 @ G3 ) )
= ( ( F3
= ( ^ [X4: A] : ( none @ B ) ) )
& ( G3
= ( ^ [X4: A] : ( none @ B ) ) ) ) ) ).
% empty_eq_map_add_iff
thf(fact_7099_map__add__upd,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B ),G3: A > ( option @ B ),X3: A,Y: B] :
( ( map_add @ A @ B @ F3 @ ( fun_upd @ A @ ( option @ B ) @ G3 @ X3 @ ( some @ B @ Y ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_add @ A @ B @ F3 @ G3 ) @ X3 @ ( some @ B @ Y ) ) ) ).
% map_add_upd
thf(fact_7100_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_7101_map__add__SomeD,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),N: B > ( option @ A ),K2: B,X3: A] :
( ( ( map_add @ B @ A @ M2 @ N @ K2 )
= ( some @ A @ X3 ) )
=> ( ( ( N @ K2 )
= ( some @ A @ X3 ) )
| ( ( ( N @ K2 )
= ( none @ A ) )
& ( ( M2 @ K2 )
= ( some @ A @ X3 ) ) ) ) ) ).
% map_add_SomeD
thf(fact_7102_map__add__Some__iff,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),N: B > ( option @ A ),K2: B,X3: A] :
( ( ( map_add @ B @ A @ M2 @ N @ K2 )
= ( some @ A @ X3 ) )
= ( ( ( N @ K2 )
= ( some @ A @ X3 ) )
| ( ( ( N @ K2 )
= ( none @ A ) )
& ( ( M2 @ K2 )
= ( some @ A @ X3 ) ) ) ) ) ).
% map_add_Some_iff
thf(fact_7103_map__add__def,axiom,
! [B: $tType,A: $tType] :
( ( map_add @ A @ B )
= ( ^ [M12: A > ( option @ B ),M23: A > ( option @ B ),X4: A] : ( case_option @ ( option @ B ) @ B @ ( M12 @ X4 ) @ ( some @ B ) @ ( M23 @ X4 ) ) ) ) ).
% map_add_def
thf(fact_7104_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_7105_map__add__upd__left,axiom,
! [A: $tType,B: $tType,M2: A,E22: A > ( option @ B ),E1: A > ( option @ B ),U1: B] :
( ~ ( member @ A @ M2 @ ( dom @ A @ B @ E22 ) )
=> ( ( map_add @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ E1 @ M2 @ ( some @ B @ U1 ) ) @ E22 )
= ( fun_upd @ A @ ( option @ B ) @ ( map_add @ A @ B @ E1 @ E22 ) @ M2 @ ( some @ B @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_7106_map__add__map__of__foldr,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),Ps: list @ ( product_prod @ A @ B )] :
( ( map_add @ A @ B @ M2 @ ( map_of @ A @ B @ Ps ) )
= ( foldr @ ( product_prod @ A @ B ) @ ( A > ( option @ B ) )
@ ( product_case_prod @ A @ B @ ( ( A > ( option @ B ) ) > A > ( option @ B ) )
@ ^ [K3: A,V5: B,M5: A > ( option @ B )] : ( fun_upd @ A @ ( option @ B ) @ M5 @ K3 @ ( some @ B @ V5 ) ) )
@ Ps
@ M2 ) ) ).
% map_add_map_of_foldr
thf(fact_7107_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_7108_drop__bit__exp__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ M2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ord_less_eq @ nat @ M2 @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% drop_bit_exp_eq
thf(fact_7109_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_7110_bit__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4730199178511100633sh_bit @ A @ M2 @ A3 ) @ N )
= ( ( ord_less_eq @ nat @ M2 @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% bit_push_bit_iff
thf(fact_7111_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_7112_bit__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( M2 = N ) ) ) ) ).
% bit_exp_iff
thf(fact_7113_bit__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ) ).
% bit_2_iff
thf(fact_7114_bit__not__exp__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( N != M2 ) ) ) ) ).
% bit_not_exp_iff
thf(fact_7115_bit__minus__exp__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% bit_minus_exp_iff
thf(fact_7116_bit__mask__sub__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% bit_mask_sub_iff
thf(fact_7117_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_7118_acyclic__insert,axiom,
! [A: $tType,Y: A,X3: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X3 ) @ R2 ) )
= ( ( transitive_acyclic @ A @ R2 )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% acyclic_insert
thf(fact_7119_set__rec,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( rec_list @ ( set @ A ) @ A @ ( bot_bot @ ( set @ A ) )
@ ^ [X4: A,Uu3: list @ A] : ( insert2 @ A @ X4 ) ) ) ).
% set_rec
thf(fact_7120_acyclicI__order,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [R2: set @ ( product_prod @ B @ B ),F3: B > A] :
( ! [A5: B,B4: B] :
( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A5 @ B4 ) @ R2 )
=> ( ord_less @ A @ ( F3 @ B4 ) @ ( F3 @ A5 ) ) )
=> ( transitive_acyclic @ B @ R2 ) ) ) ).
% acyclicI_order
thf(fact_7121_acyclicI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( transitive_acyclic @ A @ R2 ) ) ).
% acyclicI
thf(fact_7122_acyclic__def,axiom,
! [A: $tType] :
( ( transitive_acyclic @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ).
% acyclic_def
thf(fact_7123_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A] :
( ( ( semiring_gcd_Gcd_fin @ A @ A6 )
= ( zero_zero @ A ) )
= ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) )
& ( finite_finite2 @ A @ A6 ) ) ) ) ).
% Gcd_fin_0_iff
thf(fact_7124_connected__closedD,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,A6: set @ A,B5: set @ A] :
( ( topolo1966860045006549960nected @ A @ S )
=> ( ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) @ S )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
=> ( ( topolo7761053866217962861closed @ A @ A6 )
=> ( ( topolo7761053866217962861closed @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A6 @ S )
= ( bot_bot @ ( set @ A ) ) )
| ( ( inf_inf @ ( set @ A ) @ B5 @ S )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% connected_closedD
thf(fact_7125_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_7126_Gcd__fin_Oinsert,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A6: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( insert2 @ A @ A3 @ A6 ) )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ A6 ) ) ) ) ).
% Gcd_fin.insert
thf(fact_7127_connected__Times__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [S3: set @ A,T4: set @ B] :
( ( topolo1966860045006549960nected @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ S3
@ ^ [Uu3: A] : T4 ) )
= ( ( S3
= ( bot_bot @ ( set @ A ) ) )
| ( T4
= ( bot_bot @ ( set @ B ) ) )
| ( ( topolo1966860045006549960nected @ A @ S3 )
& ( topolo1966860045006549960nected @ B @ T4 ) ) ) ) ) ).
% connected_Times_eq
thf(fact_7128_gcd__list__greatest,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Bs: list @ A,A3: A] :
( ! [B4: A] :
( ( member @ A @ B4 @ ( set2 @ A @ Bs ) )
=> ( dvd_dvd @ A @ A3 @ B4 ) )
=> ( dvd_dvd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Bs ) ) ) ) ) ).
% gcd_list_greatest
thf(fact_7129_dvd__gcd__list__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,Xs2: list @ A] :
( ( dvd_dvd @ A @ B2 @ ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Xs2 ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( dvd_dvd @ A @ B2 @ X4 ) ) ) ) ) ).
% dvd_gcd_list_iff
thf(fact_7130_Gcd__fin_Osubset,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B5: set @ A,A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ( gcd_gcd @ A @ ( semiring_gcd_Gcd_fin @ A @ B5 ) @ ( semiring_gcd_Gcd_fin @ A @ A6 ) )
= ( semiring_gcd_Gcd_fin @ A @ A6 ) ) ) ) ).
% Gcd_fin.subset
thf(fact_7131_connected__contains__Ioo,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A6: set @ A,A3: A,B2: A] :
( ( topolo1966860045006549960nected @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ( member @ A @ B2 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ A6 ) ) ) ) ) ).
% connected_contains_Ioo
thf(fact_7132_connected__iff__interval,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ( ( topolo1966860045006549960nected @ A )
= ( ^ [U6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ U6 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ U6 )
=> ! [Z4: A] :
( ( ord_less_eq @ A @ X4 @ Z4 )
=> ( ( ord_less_eq @ A @ Z4 @ Y3 )
=> ( member @ A @ Z4 @ U6 ) ) ) ) ) ) ) ) ).
% connected_iff_interval
thf(fact_7133_connectedI__interval,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [U3: set @ A] :
( ! [X5: A,Y4: A,Z3: A] :
( ( member @ A @ X5 @ U3 )
=> ( ( member @ A @ Y4 @ U3 )
=> ( ( ord_less_eq @ A @ X5 @ Z3 )
=> ( ( ord_less_eq @ A @ Z3 @ Y4 )
=> ( member @ A @ Z3 @ U3 ) ) ) ) )
=> ( topolo1966860045006549960nected @ A @ U3 ) ) ) ).
% connectedI_interval
thf(fact_7134_connectedD__interval,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [U3: set @ A,X3: A,Y: A,Z2: A] :
( ( topolo1966860045006549960nected @ A @ U3 )
=> ( ( member @ A @ X3 @ U3 )
=> ( ( member @ A @ Y @ U3 )
=> ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ( ( ord_less_eq @ A @ Z2 @ Y )
=> ( member @ A @ Z2 @ U3 ) ) ) ) ) ) ) ).
% connectedD_interval
thf(fact_7135_connected__contains__Icc,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A6: set @ A,A3: A,B2: A] :
( ( topolo1966860045006549960nected @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ( member @ A @ B2 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ A6 ) ) ) ) ) ).
% connected_contains_Icc
thf(fact_7136_connected__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo1966860045006549960nected @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% connected_empty
thf(fact_7137_connected__sing,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X3: A] : ( topolo1966860045006549960nected @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% connected_sing
thf(fact_7138_connected__Un,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,T2: set @ A] :
( ( topolo1966860045006549960nected @ A @ S )
=> ( ( topolo1966860045006549960nected @ A @ T2 )
=> ( ( ( inf_inf @ ( set @ A ) @ S @ T2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( topolo1966860045006549960nected @ A @ ( sup_sup @ ( set @ A ) @ S @ T2 ) ) ) ) ) ) ).
% connected_Un
thf(fact_7139_connected__Union,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ ( set @ A )] :
( ! [S2: set @ A] :
( ( member @ ( set @ A ) @ S2 @ S3 )
=> ( topolo1966860045006549960nected @ A @ S2 ) )
=> ( ( ( complete_Inf_Inf @ ( set @ A ) @ S3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( topolo1966860045006549960nected @ A @ ( complete_Sup_Sup @ ( set @ A ) @ S3 ) ) ) ) ) ).
% connected_Union
thf(fact_7140_not__in__connected__cases,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A,X3: A] :
( ( topolo1966860045006549960nected @ A @ S3 )
=> ( ~ ( member @ A @ X3 @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( condit941137186595557371_above @ A @ S3 )
=> ~ ! [Y6: A] :
( ( member @ A @ Y6 @ S3 )
=> ( ord_less_eq @ A @ Y6 @ X3 ) ) )
=> ~ ( ( condit1013018076250108175_below @ A @ S3 )
=> ~ ! [Y6: A] :
( ( member @ A @ Y6 @ S3 )
=> ( ord_less_eq @ A @ X3 @ Y6 ) ) ) ) ) ) ) ) ).
% not_in_connected_cases
thf(fact_7141_connected__diff__open__from__closed,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,T2: set @ A,U: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ U )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( topolo7761053866217962861closed @ A @ T2 )
=> ( ( topolo1966860045006549960nected @ A @ U )
=> ( ( topolo1966860045006549960nected @ A @ ( minus_minus @ ( set @ A ) @ T2 @ S ) )
=> ( topolo1966860045006549960nected @ A @ ( minus_minus @ ( set @ A ) @ U @ S ) ) ) ) ) ) ) ) ) ).
% connected_diff_open_from_closed
thf(fact_7142_Gcd__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A6: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( insert2 @ A @ A3 @ A6 ) )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Gcd_fin.insert_remove
thf(fact_7143_Gcd__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A6: set @ A] :
( ( member @ A @ A3 @ A6 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A6 )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Gcd_fin.remove
thf(fact_7144_Gcd__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs2: list @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Xs2 ) )
= ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs2 @ ( zero_zero @ A ) ) ) ) ).
% Gcd_fin.set_eq_fold
thf(fact_7145_connectedD,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A6: set @ A,U3: set @ A,V3: set @ A] :
( ( topolo1966860045006549960nected @ A @ A6 )
=> ( ( topolo1002775350975398744n_open @ A @ U3 )
=> ( ( topolo1002775350975398744n_open @ A @ V3 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ U3 @ V3 ) @ A6 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ U3 @ V3 ) )
=> ( ( ( inf_inf @ ( set @ A ) @ U3 @ A6 )
= ( bot_bot @ ( set @ A ) ) )
| ( ( inf_inf @ ( set @ A ) @ V3 @ A6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% connectedD
thf(fact_7146_connectedI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [U3: set @ A] :
( ! [A10: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A10 )
=> ! [B7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ B7 )
=> ( ( ( inf_inf @ ( set @ A ) @ A10 @ U3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ A ) @ B7 @ U3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A10 @ B7 ) @ U3 )
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( ord_less_eq @ ( set @ A ) @ U3 @ ( sup_sup @ ( set @ A ) @ A10 @ B7 ) ) ) ) ) ) )
=> ( topolo1966860045006549960nected @ A @ U3 ) ) ) ).
% connectedI
thf(fact_7147_connected__def,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo1966860045006549960nected @ A )
= ( ^ [S6: set @ A] :
~ ? [A7: set @ A,B6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A7 )
& ( topolo1002775350975398744n_open @ A @ B6 )
& ( ord_less_eq @ ( set @ A ) @ S6 @ ( sup_sup @ ( set @ A ) @ A7 @ B6 ) )
& ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A7 @ B6 ) @ S6 )
= ( bot_bot @ ( set @ A ) ) )
& ( ( inf_inf @ ( set @ A ) @ A7 @ S6 )
!= ( bot_bot @ ( set @ A ) ) )
& ( ( inf_inf @ ( set @ A ) @ B6 @ S6 )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% connected_def
thf(fact_7148_connected__closed,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo1966860045006549960nected @ A )
= ( ^ [S7: set @ A] :
~ ? [A7: set @ A,B6: set @ A] :
( ( topolo7761053866217962861closed @ A @ A7 )
& ( topolo7761053866217962861closed @ A @ B6 )
& ( ord_less_eq @ ( set @ A ) @ S7 @ ( sup_sup @ ( set @ A ) @ A7 @ B6 ) )
& ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A7 @ B6 ) @ S7 )
= ( bot_bot @ ( set @ A ) ) )
& ( ( inf_inf @ ( set @ A ) @ A7 @ S7 )
!= ( bot_bot @ ( set @ A ) ) )
& ( ( inf_inf @ ( set @ A ) @ B6 @ S7 )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% connected_closed
thf(fact_7149_irreflp__irrefl__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irreflp @ A
@ ^ [A8: A,B8: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B8 ) @ R ) )
= ( irrefl @ A @ R ) ) ).
% irreflp_irrefl_eq
thf(fact_7150_last__list__update,axiom,
! [A: $tType,Xs2: list @ A,K2: nat,X3: A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( ( K2
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs2 @ K2 @ X3 ) )
= X3 ) )
& ( ( K2
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs2 @ K2 @ X3 ) )
= ( last @ A @ Xs2 ) ) ) ) ) ).
% last_list_update
thf(fact_7151_last__drop,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( last @ A @ ( drop @ A @ N @ Xs2 ) )
= ( last @ A @ Xs2 ) ) ) ).
% last_drop
thf(fact_7152_last__zip,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( last @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) )
= ( product_Pair @ A @ B @ ( last @ A @ Xs2 ) @ ( last @ B @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_7153_irreflpI,axiom,
! [A: $tType,R: A > A > $o] :
( ! [A5: A] :
~ ( R @ A5 @ A5 )
=> ( irreflp @ A @ R ) ) ).
% irreflpI
thf(fact_7154_irreflp__def,axiom,
! [A: $tType] :
( ( irreflp @ A )
= ( ^ [R6: A > A > $o] :
! [A8: A] :
~ ( R6 @ A8 @ A8 ) ) ) ).
% irreflp_def
thf(fact_7155_last__in__set,axiom,
! [A: $tType,As2: list @ A] :
( ( As2
!= ( nil @ A ) )
=> ( member @ A @ ( last @ A @ As2 ) @ ( set2 @ A @ As2 ) ) ) ).
% last_in_set
thf(fact_7156_irreflp__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( irreflp @ A @ ( ord_less @ A ) ) ) ).
% irreflp_less
thf(fact_7157_dropWhile__last,axiom,
! [A: $tType,X3: A,Xs2: list @ A,P: A > $o] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ~ ( P @ X3 )
=> ( ( last @ A @ ( dropWhile @ A @ P @ Xs2 ) )
= ( last @ A @ Xs2 ) ) ) ) ).
% dropWhile_last
thf(fact_7158_irreflp__greater,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( irreflp @ A
@ ^ [X4: A,Y3: A] : ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% irreflp_greater
thf(fact_7159_last__conv__nth,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( last @ A @ Xs2 )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) ) ) ) ).
% last_conv_nth
thf(fact_7160_minus__coset__filter,axiom,
! [A: $tType,A6: set @ A,Xs2: list @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ ( coset @ A @ Xs2 ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A6 )
@ Xs2 ) ) ) ).
% minus_coset_filter
thf(fact_7161_disjnt__equiv__class,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( disjnt @ A @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_7162_disjnt__self__iff__empty,axiom,
! [A: $tType,S3: set @ A] :
( ( disjnt @ A @ S3 @ S3 )
= ( S3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjnt_self_iff_empty
thf(fact_7163_disjnt__insert2,axiom,
! [A: $tType,Y8: set @ A,A3: A,X6: set @ A] :
( ( disjnt @ A @ Y8 @ ( insert2 @ A @ A3 @ X6 ) )
= ( ~ ( member @ A @ A3 @ Y8 )
& ( disjnt @ A @ Y8 @ X6 ) ) ) ).
% disjnt_insert2
thf(fact_7164_disjnt__insert1,axiom,
! [A: $tType,A3: A,X6: set @ A,Y8: set @ A] :
( ( disjnt @ A @ ( insert2 @ A @ A3 @ X6 ) @ Y8 )
= ( ~ ( member @ A @ A3 @ Y8 )
& ( disjnt @ A @ X6 @ Y8 ) ) ) ).
% disjnt_insert1
thf(fact_7165_disjnt__Un1,axiom,
! [A: $tType,A6: set @ A,B5: set @ A,C4: set @ A] :
( ( disjnt @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) @ C4 )
= ( ( disjnt @ A @ A6 @ C4 )
& ( disjnt @ A @ B5 @ C4 ) ) ) ).
% disjnt_Un1
thf(fact_7166_disjnt__Un2,axiom,
! [A: $tType,C4: set @ A,A6: set @ A,B5: set @ A] :
( ( disjnt @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( ( disjnt @ A @ C4 @ A6 )
& ( disjnt @ A @ C4 @ B5 ) ) ) ).
% disjnt_Un2
thf(fact_7167_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C4: set @ A,A6: set @ B,B5: set @ B] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ C4
@ ^ [Uu3: A] : A6 )
@ ( product_Sigma @ A @ B @ C4
@ ^ [Uu3: A] : B5 ) )
= ( ( C4
= ( bot_bot @ ( set @ A ) ) )
| ( disjnt @ B @ A6 @ B5 ) ) ) ).
% disjnt_Times1_iff
thf(fact_7168_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A6: set @ A,C4: set @ B,B5: set @ A] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : C4 )
@ ( product_Sigma @ A @ B @ B5
@ ^ [Uu3: A] : C4 ) )
= ( ( C4
= ( bot_bot @ ( set @ B ) ) )
| ( disjnt @ A @ A6 @ B5 ) ) ) ).
% disjnt_Times2_iff
thf(fact_7169_disjnt__sym,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( disjnt @ A @ A6 @ B5 )
=> ( disjnt @ A @ B5 @ A6 ) ) ).
% disjnt_sym
thf(fact_7170_disjnt__iff,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A7: set @ A,B6: set @ A] :
! [X4: A] :
~ ( ( member @ A @ X4 @ A7 )
& ( member @ A @ X4 @ B6 ) ) ) ) ).
% disjnt_iff
thf(fact_7171_disjnt__subset2,axiom,
! [A: $tType,X6: set @ A,Y8: set @ A,Z7: set @ A] :
( ( disjnt @ A @ X6 @ Y8 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z7 @ Y8 )
=> ( disjnt @ A @ X6 @ Z7 ) ) ) ).
% disjnt_subset2
thf(fact_7172_disjnt__subset1,axiom,
! [A: $tType,X6: set @ A,Y8: set @ A,Z7: set @ A] :
( ( disjnt @ A @ X6 @ Y8 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z7 @ X6 )
=> ( disjnt @ A @ Z7 @ Y8 ) ) ) ).
% disjnt_subset1
thf(fact_7173_disjnt__def,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A7 @ B6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjnt_def
thf(fact_7174_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A6: set @ A,C4: A > ( set @ B ),B5: set @ A] :
( ( disjnt @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A6 @ C4 ) @ ( product_Sigma @ A @ B @ B5 @ C4 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) )
=> ( ( C4 @ X4 )
= ( bot_bot @ ( set @ B ) ) ) )
| ( disjnt @ A @ A6 @ B5 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_7175_disjnt__empty2,axiom,
! [A: $tType,A6: set @ A] : ( disjnt @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ).
% disjnt_empty2
thf(fact_7176_disjnt__empty1,axiom,
! [A: $tType,A6: set @ A] : ( disjnt @ A @ ( bot_bot @ ( set @ A ) ) @ A6 ) ).
% disjnt_empty1
thf(fact_7177_disjnt__insert,axiom,
! [A: $tType,X3: A,N5: set @ A,M7: set @ A] :
( ~ ( member @ A @ X3 @ N5 )
=> ( ( disjnt @ A @ M7 @ N5 )
=> ( disjnt @ A @ ( insert2 @ A @ X3 @ M7 ) @ N5 ) ) ) ).
% disjnt_insert
thf(fact_7178_subset__code_I2_J,axiom,
! [B: $tType,A6: set @ B,Ys: list @ B] :
( ( ord_less_eq @ ( set @ B ) @ A6 @ ( coset @ B @ Ys ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Ys ) )
=> ~ ( member @ B @ X4 @ A6 ) ) ) ) ).
% subset_code(2)
thf(fact_7179_compl__coset,axiom,
! [A: $tType,Xs2: list @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( coset @ A @ Xs2 ) )
= ( set2 @ A @ Xs2 ) ) ).
% compl_coset
thf(fact_7180_coset__def,axiom,
! [A: $tType] :
( ( coset @ A )
= ( ^ [Xs: list @ A] : ( uminus_uminus @ ( set @ A ) @ ( set2 @ A @ Xs ) ) ) ) ).
% coset_def
thf(fact_7181_insert__code_I2_J,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( insert2 @ A @ X3 @ ( coset @ A @ Xs2 ) )
= ( coset @ A @ ( removeAll @ A @ X3 @ Xs2 ) ) ) ).
% insert_code(2)
thf(fact_7182_card__Un__disjnt,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( disjnt @ A @ A6 @ B5 )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ) ).
% card_Un_disjnt
thf(fact_7183_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_7184_infinite__infinite__partition,axiom,
! [A: $tType,A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ~ ! [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 ) ) ) ) @ A6 )
=> ~ ! [I2: nat] :
~ ( finite_finite2 @ A @ ( C7 @ I2 ) ) ) ) ) ).
% infinite_infinite_partition
thf(fact_7185_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 ),S7: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ S7 )
& ! [A8: A,B8: A,C6: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B8 ) @ S7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ C6 ) @ R5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B8 ) @ R5 ) ) ) ) ) ) ).
% init_seg_of_def
thf(fact_7186_pairwise__insert,axiom,
! [A: $tType,R2: A > A > $o,X3: A,S: set @ A] :
( ( pairwise @ A @ R2 @ ( insert2 @ A @ X3 @ S ) )
= ( ! [Y3: A] :
( ( ( member @ A @ Y3 @ S )
& ( Y3 != X3 ) )
=> ( ( R2 @ X3 @ Y3 )
& ( R2 @ Y3 @ X3 ) ) )
& ( pairwise @ A @ R2 @ S ) ) ) ).
% pairwise_insert
thf(fact_7187_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_7188_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A6: A] : ( pairwise @ A @ P @ ( insert2 @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_7189_pairwise__imageI,axiom,
! [B: $tType,A: $tType,A6: set @ A,F3: A > B,P: B > B > $o] :
( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ A6 )
=> ( ( member @ A @ Y4 @ A6 )
=> ( ( X5 != Y4 )
=> ( ( ( F3 @ X5 )
!= ( F3 @ Y4 ) )
=> ( P @ ( F3 @ X5 ) @ ( F3 @ Y4 ) ) ) ) ) )
=> ( pairwise @ B @ P @ ( image2 @ A @ B @ F3 @ A6 ) ) ) ).
% pairwise_imageI
thf(fact_7190_pairwise__image,axiom,
! [A: $tType,B: $tType,R2: A > A > $o,F3: B > A,S: set @ B] :
( ( pairwise @ A @ R2 @ ( image2 @ B @ A @ F3 @ S ) )
= ( pairwise @ B
@ ^ [X4: B,Y3: B] :
( ( ( F3 @ X4 )
!= ( F3 @ Y3 ) )
=> ( R2 @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
@ S ) ) ).
% pairwise_image
thf(fact_7191_pairwise__mono,axiom,
! [A: $tType,P: A > A > $o,A6: set @ A,Q: A > A > $o,B5: set @ A] :
( ( pairwise @ A @ P @ A6 )
=> ( ! [X5: A,Y4: A] :
( ( P @ X5 @ Y4 )
=> ( Q @ X5 @ Y4 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( pairwise @ A @ Q @ B5 ) ) ) ) ).
% pairwise_mono
thf(fact_7192_pairwise__subset,axiom,
! [A: $tType,P: A > A > $o,S3: set @ A,T4: set @ A] :
( ( pairwise @ A @ P @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T4 @ S3 )
=> ( pairwise @ A @ P @ T4 ) ) ) ).
% pairwise_subset
thf(fact_7193_refl__on__init__seg__of,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 @ R2 ) @ ( init_seg_of @ A ) ) ).
% refl_on_init_seg_of
thf(fact_7194_antisym__init__seg__of,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: 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 @ S ) @ ( init_seg_of @ 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 ) ) @ S @ R2 ) @ ( init_seg_of @ A ) )
=> ( R2 = S ) ) ) ).
% antisym_init_seg_of
thf(fact_7195_trans__init__seg__of,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),T2: 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 @ S ) @ ( init_seg_of @ 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 ) ) @ S @ T2 ) @ ( init_seg_of @ 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 @ T2 ) @ ( init_seg_of @ A ) ) ) ) ).
% trans_init_seg_of
thf(fact_7196_pairwiseD,axiom,
! [A: $tType,R: A > A > $o,S3: set @ A,X3: A,Y: A] :
( ( pairwise @ A @ R @ S3 )
=> ( ( member @ A @ X3 @ S3 )
=> ( ( member @ A @ Y @ S3 )
=> ( ( X3 != Y )
=> ( R @ X3 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_7197_pairwiseI,axiom,
! [A: $tType,S3: set @ A,R: A > A > $o] :
( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ S3 )
=> ( ( member @ A @ Y4 @ S3 )
=> ( ( X5 != Y4 )
=> ( R @ X5 @ Y4 ) ) ) )
=> ( pairwise @ A @ R @ S3 ) ) ).
% pairwiseI
thf(fact_7198_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R6: A > A > $o,S6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ S6 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ S6 )
=> ( ( X4 != Y3 )
=> ( R6 @ X4 @ Y3 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_7199_pairwise__trivial,axiom,
! [A: $tType,I5: set @ A] :
( pairwise @ A
@ ^ [I4: A,J3: A] : J3 != I4
@ I5 ) ).
% pairwise_trivial
thf(fact_7200_initial__segment__of__Diff,axiom,
! [A: $tType,P2: set @ ( product_prod @ A @ A ),Q3: set @ ( product_prod @ A @ A ),S: 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 ) ) @ P2 @ Q3 ) @ ( init_seg_of @ 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 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ P2 @ S ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ Q3 @ S ) ) @ ( init_seg_of @ A ) ) ) ).
% initial_segment_of_Diff
thf(fact_7201_pairwise__alt,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R6: A > A > $o,S6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ S6 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ ( minus_minus @ ( set @ A ) @ S6 @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( R6 @ X4 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_7202_Chains__init__seg__of__Union,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) ),R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) @ R @ ( chains @ ( set @ ( product_prod @ A @ A ) ) @ ( init_seg_of @ A ) ) )
=> ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R )
=> ( 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 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R ) ) @ ( init_seg_of @ A ) ) ) ) ).
% Chains_init_seg_of_Union
thf(fact_7203_insort__insert__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Xs2: list @ B] :
( ~ ( member @ A @ ( F3 @ X3 ) @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs2 ) ) )
=> ( ( linord329482645794927042rt_key @ B @ A @ F3 @ X3 @ Xs2 )
= ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs2 ) ) ) ) ).
% insort_insert_insort_key
thf(fact_7204_insort__insert__triv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ Xs2 )
= Xs2 ) ) ) ).
% insort_insert_triv
thf(fact_7205_insort__insert__key__triv,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Xs2: list @ B] :
( ( member @ A @ ( F3 @ X3 ) @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs2 ) ) )
=> ( ( linord329482645794927042rt_key @ B @ A @ F3 @ X3 @ Xs2 )
= Xs2 ) ) ) ).
% insort_insert_key_triv
thf(fact_7206_disjoint__image__subset,axiom,
! [A: $tType,A20: set @ ( set @ A ),F3: ( set @ A ) > ( set @ A )] :
( ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ A20 )
=> ( ! [X10: set @ A] :
( ( member @ ( set @ A ) @ X10 @ A20 )
=> ( ord_less_eq @ ( set @ A ) @ ( F3 @ X10 ) @ X10 ) )
=> ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ F3 @ A20 ) ) ) ) ).
% disjoint_image_subset
thf(fact_7207_set__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Xs2: list @ A] :
( ( set2 @ A
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ Xs2 ) )
= ( insert2 @ A @ X3 @ ( set2 @ A @ Xs2 ) ) ) ) ).
% set_insort_insert
thf(fact_7208_sorted__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,X3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ Xs2 ) ) ) ) ).
% sorted_insort_insert
thf(fact_7209_Chains__def,axiom,
! [A: $tType] :
( ( chains @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( set @ A )
@ ^ [C5: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ C5 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ C5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% Chains_def
thf(fact_7210_Chains__relation__of,axiom,
! [A: $tType,C4: set @ A,P: A > A > $o,A6: set @ A] :
( ( member @ ( set @ A ) @ C4 @ ( chains @ A @ ( order_relation_of @ A @ P @ A6 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ C4 @ A6 ) ) ).
% Chains_relation_of
thf(fact_7211_insort__insert__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Xs2: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ Xs2 )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ Xs2 ) ) ) ) ).
% insort_insert_insort
thf(fact_7212_sorted__insort__insert__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs2: list @ B,X3: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs2 ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( linord329482645794927042rt_key @ B @ A @ F3 @ X3 @ Xs2 ) ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_7213_insort__insert__key__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linord329482645794927042rt_key @ B @ A )
= ( ^ [F4: B > A,X4: B,Xs: list @ B] : ( if @ ( list @ B ) @ ( member @ A @ ( F4 @ X4 ) @ ( image2 @ B @ A @ F4 @ ( set2 @ B @ Xs ) ) ) @ Xs @ ( linorder_insort_key @ B @ A @ F4 @ X4 @ Xs ) ) ) ) ) ).
% insort_insert_key_def
thf(fact_7214_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 ) )
@ ^ [X4: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 ) ) ) ) ).
% Chains_subset
thf(fact_7215_sort__key__conv__fold,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs2: list @ B] :
( ( inj_on @ B @ A @ F3 @ ( set2 @ B @ Xs2 ) )
=> ( ( linorder_sort_key @ B @ A @ F3 @ Xs2 )
= ( fold @ B @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F3 ) @ Xs2 @ ( nil @ B ) ) ) ) ) ).
% sort_key_conv_fold
thf(fact_7216_set__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs2: list @ B] :
( ( set2 @ B @ ( linorder_sort_key @ B @ A @ F3 @ Xs2 ) )
= ( set2 @ B @ Xs2 ) ) ) ).
% set_sort
thf(fact_7217_length__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs2: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_sort_key @ B @ A @ F3 @ Xs2 ) )
= ( size_size @ ( list @ B ) @ Xs2 ) ) ) ).
% length_sort
thf(fact_7218_pred__on_Ochain__empty,axiom,
! [A: $tType,A6: set @ A,P: A > A > $o] : ( pred_chain @ A @ A6 @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pred_on.chain_empty
thf(fact_7219_pred__on_Ochain__def,axiom,
! [A: $tType] :
( ( pred_chain @ A )
= ( ^ [A7: set @ A,P4: A > A > $o,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ A7 )
& ! [X4: A] :
( ( member @ A @ X4 @ C5 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ C5 )
=> ( ( sup_sup @ ( A > A > $o ) @ P4
@ ^ [Y5: A,Z: A] : Y5 = Z
@ X4
@ Y3 )
| ( sup_sup @ ( A > A > $o ) @ P4
@ ^ [Y5: A,Z: A] : Y5 = Z
@ Y3
@ X4 ) ) ) ) ) ) ) ).
% pred_on.chain_def
thf(fact_7220_pred__on_OchainI,axiom,
! [A: $tType,C4: set @ A,A6: set @ A,P: A > A > $o] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ A6 )
=> ( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ C4 )
=> ( ( member @ A @ Y4 @ C4 )
=> ( ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z: A] : Y5 = Z
@ X5
@ Y4 )
| ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z: A] : Y5 = Z
@ Y4
@ X5 ) ) ) )
=> ( pred_chain @ A @ A6 @ P @ C4 ) ) ) ).
% pred_on.chainI
thf(fact_7221_subset__Zorn,axiom,
! [A: $tType,A6: set @ ( set @ A )] :
( ! [C7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C7 )
=> ? [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A6 )
& ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ C7 )
=> ( ord_less_eq @ ( set @ A ) @ Xa3 @ X ) ) ) )
=> ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A6 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ Xa )
=> ( Xa = X5 ) ) ) ) ) ).
% subset_Zorn
thf(fact_7222_subset__Zorn_H,axiom,
! [A: $tType,A6: set @ ( set @ A )] :
( ! [C7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C7 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C7 ) @ A6 ) )
=> ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A6 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ Xa )
=> ( Xa = X5 ) ) ) ) ) ).
% subset_Zorn'
thf(fact_7223_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 )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C10 )
=> ! [Y3: set @ A] :
( ( member @ ( set @ A ) @ Y3 @ C10 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ Y3 )
| ( ord_less_eq @ ( set @ A ) @ Y3 @ X4 ) ) ) ) ) ) ).
% subset_chain_def
thf(fact_7224_subset__chain__insert,axiom,
! [A: $tType,A20: set @ ( set @ A ),B5: set @ A,B11: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ ( insert2 @ ( set @ A ) @ B5 @ B11 ) )
= ( ( member @ ( set @ A ) @ B5 @ A20 )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ B11 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ B5 )
| ( ord_less_eq @ ( set @ A ) @ B5 @ X4 ) ) )
& ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ B11 ) ) ) ).
% subset_chain_insert
thf(fact_7225_sorted__sort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs2 ) ) ) ).
% sorted_sort
thf(fact_7226_sorted__sort__id,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs2 )
= Xs2 ) ) ) ).
% sorted_sort_id
thf(fact_7227_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 ) ) )
=> ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A20 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A20 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ Xa )
=> ( Xa = X5 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_7228_sorted__sort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs2: list @ B] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( linorder_sort_key @ B @ A @ F3 @ Xs2 ) ) ) ) ).
% sorted_sort_key
thf(fact_7229_pred__on_Ochain__extend,axiom,
! [A: $tType,A6: set @ A,P: A > A > $o,C4: set @ A,Z2: A] :
( ( pred_chain @ A @ A6 @ P @ C4 )
=> ( ( member @ A @ Z2 @ A6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ C4 )
=> ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z: A] : Y5 = Z
@ X5
@ Z2 ) )
=> ( pred_chain @ A @ A6 @ P @ ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) @ C4 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_7230_finite__subset__Union__chain,axiom,
! [A: $tType,A6: set @ A,B11: set @ ( set @ A ),A20: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) )
=> ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ B11 )
=> ~ ! [B7: set @ A] :
( ( member @ ( set @ A ) @ B7 @ B11 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A6 @ B7 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_7231_sorted__list__of__set__sort__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs2 ) )
= ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ ( remdups @ A @ Xs2 ) ) ) ) ).
% sorted_list_of_set_sort_remdups
thf(fact_7232_Bleast__code,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,P: A > $o] :
( ( bleast @ A @ ( set2 @ A @ Xs2 ) @ P )
= ( case_list @ A @ A @ ( abort_Bleast @ A @ ( set2 @ A @ Xs2 ) @ P )
@ ^ [X4: A,Xs: list @ A] : X4
@ ( filter2 @ A @ P
@ ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs2 ) ) ) ) ) ).
% Bleast_code
thf(fact_7233_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 ) )
@ ^ [X4: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 ) ) )
@ ( chains @ A @ R2 ) ) ) ).
% Chains_subset'
thf(fact_7234_refl__Id,axiom,
! [A: $tType] : ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ ( id2 @ A ) ) ).
% refl_Id
thf(fact_7235_refl__on__Un,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),B5: set @ A,S: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A6 @ R2 )
=> ( ( refl_on @ A @ B5 @ S )
=> ( refl_on @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S ) ) ) ) ).
% refl_on_Un
thf(fact_7236_refl__on__Id__on,axiom,
! [A: $tType,A6: set @ A] : ( refl_on @ A @ A6 @ ( id_on @ A @ A6 ) ) ).
% refl_on_Id_on
thf(fact_7237_refl__onD2,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ( refl_on @ A @ A6 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( member @ A @ Y @ A6 ) ) ) ).
% refl_onD2
thf(fact_7238_refl__onD1,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ( refl_on @ A @ A6 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( member @ A @ X3 @ A6 ) ) ) ).
% refl_onD1
thf(fact_7239_refl__onD,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( refl_on @ A @ A6 @ R2 )
=> ( ( member @ A @ A3 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_7240_refl__on__domain,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( refl_on @ A @ A6 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( ( member @ A @ A3 @ A6 )
& ( member @ A @ B2 @ A6 ) ) ) ) ).
% refl_on_domain
thf(fact_7241_refl__on__Int,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),B5: set @ A,S: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A6 @ R2 )
=> ( ( refl_on @ A @ B5 @ S )
=> ( refl_on @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B5 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S ) ) ) ) ).
% refl_on_Int
thf(fact_7242_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_7243_refl__onI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu3: A] : A6 ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R2 ) )
=> ( refl_on @ A @ A6 @ R2 ) ) ) ).
% refl_onI
thf(fact_7244_refl__on__def,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ A7
@ ^ [Uu3: A] : A7 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R5 ) ) ) ) ) ).
% refl_on_def
thf(fact_7245_refl__on__def_H,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Z4: A] :
( ( member @ A @ Y3 @ A7 )
& ( member @ A @ Z4 @ A7 ) )
@ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R5 ) ) ) ) ) ).
% refl_on_def'
thf(fact_7246_refl__on__UNION,axiom,
! [B: $tType,A: $tType,S3: set @ A,A6: A > ( set @ B ),R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( refl_on @ B @ ( A6 @ X5 ) @ ( R2 @ X5 ) ) )
=> ( refl_on @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ S3 ) ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S3 ) ) ) ) ).
% refl_on_UNION
thf(fact_7247_Refl__Field__Restr2,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( field2 @ A @ R2 ) )
=> ( ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu3: A] : A6 ) ) )
= A6 ) ) ) ).
% Refl_Field_Restr2
thf(fact_7248_refl__on__INTER,axiom,
! [B: $tType,A: $tType,S3: set @ A,A6: A > ( set @ B ),R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( refl_on @ B @ ( A6 @ X5 ) @ ( R2 @ X5 ) ) )
=> ( refl_on @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ S3 ) ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S3 ) ) ) ) ).
% refl_on_INTER
thf(fact_7249_Chains__alt__def,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R2 )
=> ( ( chains @ A @ R2 )
= ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X4: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 ) ) ) ) ) ).
% Chains_alt_def
thf(fact_7250_refl__on__singleton,axiom,
! [A: $tType,X3: A] : ( refl_on @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% refl_on_singleton
thf(fact_7251_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_7252_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( antisym @ A @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A3 = B2 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_7253_antisym__empty,axiom,
! [A: $tType] : ( antisym @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% antisym_empty
thf(fact_7254_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_7255_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_7256_antisym__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
=> ( ( antisym @ A @ S )
=> ( antisym @ A @ R2 ) ) ) ).
% antisym_subset
thf(fact_7257_under__def,axiom,
! [A: $tType] :
( ( order_under @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A8: A] :
( collect @ A
@ ^ [B8: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ A8 ) @ R5 ) ) ) ) ).
% under_def
thf(fact_7258_antisymD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( antisym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R2 )
=> ( A3 = B2 ) ) ) ) ).
% antisymD
thf(fact_7259_antisymI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X5: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X5 ) @ R2 )
=> ( X5 = Y4 ) ) )
=> ( antisym @ A @ R2 ) ) ).
% antisymI
thf(fact_7260_antisym__def,axiom,
! [A: $tType] :
( ( antisym @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R5 )
=> ( X4 = Y3 ) ) ) ) ) ).
% antisym_def
thf(fact_7261_antisym__Id,axiom,
! [A: $tType] : ( antisym @ A @ ( id2 @ A ) ) ).
% antisym_Id
thf(fact_7262_antisym__Id__on,axiom,
! [A: $tType,A6: set @ A] : ( antisym @ A @ ( id_on @ A @ A6 ) ) ).
% antisym_Id_on
thf(fact_7263_antisym__singleton,axiom,
! [A: $tType,X3: product_prod @ A @ A] : ( antisym @ A @ ( insert2 @ ( product_prod @ A @ A ) @ X3 @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% antisym_singleton
thf(fact_7264_sqr_Osimps_I3_J,axiom,
! [N: num] :
( ( sqr @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ ( plus_plus @ num @ ( sqr @ N ) @ N ) ) ) ) ).
% sqr.simps(3)
thf(fact_7265_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 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ A3 ) @ ( F3 @ Y4 ) ) )
=> ( ( lattices_ord_arg_min @ A @ B @ F3 @ P )
= A3 ) ) ) ) ) ).
% arg_min_inj_eq
thf(fact_7266_sqr_Osimps_I1_J,axiom,
( ( sqr @ one2 )
= one2 ) ).
% sqr.simps(1)
thf(fact_7267_sqr_Osimps_I2_J,axiom,
! [N: num] :
( ( sqr @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% sqr.simps(2)
thf(fact_7268_arg__min__nat__lemma,axiom,
! [A: $tType,P: A > $o,K2: A,M2: A > nat] :
( ( P @ K2 )
=> ( ( P @ ( lattices_ord_arg_min @ A @ nat @ M2 @ P ) )
& ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ nat @ ( M2 @ ( lattices_ord_arg_min @ A @ nat @ M2 @ P ) ) @ ( M2 @ Y6 ) ) ) ) ) ).
% arg_min_nat_lemma
thf(fact_7269_arg__min__nat__le,axiom,
! [A: $tType,P: A > $o,X3: A,M2: A > nat] :
( ( P @ X3 )
=> ( ord_less_eq @ nat @ ( M2 @ ( lattices_ord_arg_min @ A @ nat @ M2 @ P ) ) @ ( M2 @ X3 ) ) ) ).
% arg_min_nat_le
thf(fact_7270_arg__min__equality,axiom,
! [A: $tType,C: $tType] :
( ( order @ A )
=> ! [P: C > $o,K2: C,F3: C > A] :
( ( P @ K2 )
=> ( ! [X5: C] :
( ( P @ X5 )
=> ( ord_less_eq @ A @ ( F3 @ K2 ) @ ( F3 @ X5 ) ) )
=> ( ( F3 @ ( lattices_ord_arg_min @ C @ A @ F3 @ P ) )
= ( F3 @ K2 ) ) ) ) ) ).
% arg_min_equality
thf(fact_7271_numeral__sqr,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K2: num] :
( ( numeral_numeral @ A @ ( sqr @ K2 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ K2 ) @ ( numeral_numeral @ A @ K2 ) ) ) ) ).
% numeral_sqr
thf(fact_7272_pow_Osimps_I2_J,axiom,
! [X3: num,Y: num] :
( ( pow @ X3 @ ( bit0 @ Y ) )
= ( sqr @ ( pow @ X3 @ Y ) ) ) ).
% pow.simps(2)
thf(fact_7273_Rep__unit__induct,axiom,
! [Y: $o,P: $o > $o] :
( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ! [X5: product_unit] : ( P @ ( product_Rep_unit @ X5 ) )
=> ( P @ Y ) ) ) ).
% Rep_unit_induct
thf(fact_7274_Abs__unit__inject,axiom,
! [X3: $o,Y: $o] :
( ( member @ $o @ X3 @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( ( product_Abs_unit @ X3 )
= ( product_Abs_unit @ Y ) )
= ( X3 = Y ) ) ) ) ).
% Abs_unit_inject
thf(fact_7275_Abs__unit__inverse,axiom,
! [Y: $o] :
( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( product_Rep_unit @ ( product_Abs_unit @ Y ) )
= Y ) ) ).
% Abs_unit_inverse
thf(fact_7276_Rep__unit__inject,axiom,
! [X3: product_unit,Y: product_unit] :
( ( ( product_Rep_unit @ X3 )
= ( product_Rep_unit @ Y ) )
= ( X3 = Y ) ) ).
% Rep_unit_inject
thf(fact_7277_Rep__unit__inverse,axiom,
! [X3: product_unit] :
( ( product_Abs_unit @ ( product_Rep_unit @ X3 ) )
= X3 ) ).
% Rep_unit_inverse
thf(fact_7278_Rep__unit,axiom,
! [X3: product_unit] : ( member @ $o @ ( product_Rep_unit @ X3 ) @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ).
% Rep_unit
thf(fact_7279_Abs__unit__cases,axiom,
! [X3: product_unit] :
~ ! [Y4: $o] :
( ( X3
= ( product_Abs_unit @ Y4 ) )
=> ~ ( member @ $o @ Y4 @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% Abs_unit_cases
thf(fact_7280_Rep__unit__cases,axiom,
! [Y: $o] :
( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ~ ! [X5: product_unit] :
( Y
= ( ~ ( product_Rep_unit @ X5 ) ) ) ) ).
% Rep_unit_cases
thf(fact_7281_Abs__unit__induct,axiom,
! [P: product_unit > $o,X3: product_unit] :
( ! [Y4: $o] :
( ( member @ $o @ Y4 @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( P @ ( product_Abs_unit @ Y4 ) ) )
=> ( P @ X3 ) ) ).
% Abs_unit_induct
thf(fact_7282_type__definition__unit,axiom,
type_definition @ product_unit @ $o @ product_Rep_unit @ product_Abs_unit @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ).
% type_definition_unit
thf(fact_7283_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_7284_in__measures_I2_J,axiom,
! [A: $tType,X3: A,Y: A,F3: A > nat,Fs: list @ ( A > nat )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) )
= ( ( ord_less @ nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X3 )
= ( F3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_7285_Partial__order__eq__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A3 = B2 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_7286_in__measures_I1_J,axiom,
! [A: $tType,X3: A,Y: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measures @ A @ ( nil @ ( A > nat ) ) ) ) ).
% in_measures(1)
thf(fact_7287_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_7288_measures__less,axiom,
! [A: $tType,F3: A > nat,X3: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less @ nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) ) ) ).
% measures_less
thf(fact_7289_measures__lesseq,axiom,
! [A: $tType,F3: A > nat,X3: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less_eq @ nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measures @ A @ Fs ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_7290_Zorns__po__lemma,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ! [C7: set @ A] :
( ( member @ ( set @ A ) @ C7 @ ( chains @ A @ R2 ) )
=> ? [X: A] :
( ( member @ A @ X @ ( field2 @ A @ R2 ) )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ C7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa3 @ X ) @ R2 ) ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R2 ) )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa ) @ R2 )
=> ( Xa = X5 ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_7291_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) )
=> ( ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( A3 = B2 )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_7292_zero__rat__def,axiom,
( ( zero_zero @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ) ).
% zero_rat_def
thf(fact_7293_wo__rel_Omax2__greater,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B2 ) ) @ R2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B2 ) ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_7294_wo__rel_Omax2__equals2,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B2 )
= B2 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_7295_wo__rel_Omax2__equals1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B2 )
= A3 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_7296_wo__rel_Omax2__def,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B2 )
= B2 ) )
& ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B2 )
= A3 ) ) ) ) ).
% wo_rel.max2_def
thf(fact_7297_wo__rel_Owell__order__induct,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ! [X5: A] :
( ! [Y6: A] :
( ( ( Y6 != X5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X5 ) @ R2 ) )
=> ( P @ Y6 ) )
=> ( P @ X5 ) )
=> ( P @ A3 ) ) ) ).
% wo_rel.well_order_induct
thf(fact_7298_wo__rel_OTOTALS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ! [X: A] :
( ( member @ A @ X @ ( field2 @ A @ R2 ) )
=> ! [Xa: A] :
( ( member @ A @ Xa @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Xa ) @ R2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa @ X ) @ R2 ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_7299_well__order__induct__imp,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ! [X5: A] :
( ! [Y6: A] :
( ( ( Y6 != X5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X5 ) @ R2 ) )
=> ( ( member @ A @ Y6 @ ( field2 @ A @ R2 ) )
=> ( P @ Y6 ) ) )
=> ( ( member @ A @ X5 @ ( field2 @ A @ R2 ) )
=> ( P @ X5 ) ) )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( P @ A3 ) ) ) ) ).
% well_order_induct_imp
thf(fact_7300_wo__rel_Omax2__among,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R2 ) )
=> ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B2 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_7301_wo__rel_Ocases__Total,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R2 )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_7302_natLeq__on__wo__rel,axiom,
! [N: nat] :
( bNF_Wellorder_wo_rel @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y3: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X4 @ Y3 ) ) ) ) ) ).
% natLeq_on_wo_rel
thf(fact_7303_wo__rel_Omax2__greater__among,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B2 ) ) @ R2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B2 ) ) @ R2 )
& ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B2 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_7304_one__rat__def,axiom,
( ( one_one @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ) ).
% one_rat_def
thf(fact_7305_wo__rel_OWell__order__isMinim__exists,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_12: A] : ( bNF_We4791949203932849705sMinim @ A @ R2 @ B5 @ X_12 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_7306_plus__rat_Oabs__eq,axiom,
! [Xa2: product_prod @ int @ int,X3: product_prod @ int @ int] :
( ( ratrel @ Xa2 @ Xa2 )
=> ( ( ratrel @ X3 @ X3 )
=> ( ( plus_plus @ rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X3 ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa2 ) @ ( product_snd @ int @ int @ X3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa2 ) @ ( product_snd @ int @ int @ X3 ) ) ) ) ) ) ) ).
% plus_rat.abs_eq
thf(fact_7307_one__rat_Orsp,axiom,
ratrel @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ).
% one_rat.rsp
thf(fact_7308_zero__rat_Orsp,axiom,
ratrel @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ).
% zero_rat.rsp
thf(fact_7309_wo__rel_OisMinim__def,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( bNF_We4791949203932849705sMinim @ A @ R2 @ A6 @ B2 )
= ( ( member @ A @ B2 @ A6 )
& ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ X4 ) @ R2 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_7310_uminus__rat_Oabs__eq,axiom,
! [X3: product_prod @ int @ int] :
( ( ratrel @ X3 @ X3 )
=> ( ( uminus_uminus @ rat @ ( abs_Rat @ X3 ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) ) ) ) ) ).
% uminus_rat.abs_eq
thf(fact_7311_times__rat_Oabs__eq,axiom,
! [Xa2: product_prod @ int @ int,X3: product_prod @ int @ int] :
( ( ratrel @ Xa2 @ Xa2 )
=> ( ( ratrel @ X3 @ X3 )
=> ( ( times_times @ rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X3 ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa2 ) @ ( product_fst @ int @ int @ X3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa2 ) @ ( product_snd @ int @ int @ X3 ) ) ) ) ) ) ) ).
% times_rat.abs_eq
thf(fact_7312_inverse__rat_Oabs__eq,axiom,
! [X3: product_prod @ int @ int] :
( ( ratrel @ X3 @ X3 )
=> ( ( inverse_inverse @ rat @ ( abs_Rat @ X3 ) )
= ( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X3 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) ) ) ) ) ).
% inverse_rat.abs_eq
thf(fact_7313_wo__rel_Ominim__isMinim,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( bNF_We4791949203932849705sMinim @ A @ R2 @ B5 @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B5 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_7314_wo__rel_Ominim__in,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B5 ) @ B5 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_7315_wo__rel_Ominim__least,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B2 @ B5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B5 ) @ B2 ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_7316_wo__rel_Oequals__minim,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ A3 @ B5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ B5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B4 ) @ R2 ) )
=> ( A3
= ( bNF_We6954850376910717587_minim @ A @ R2 @ B5 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_7317_wo__rel_Ominim__inField,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B5 ) @ ( field2 @ A @ R2 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_7318_inverse__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
@ ^ [X4: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X4 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_fst @ int @ int @ X4 ) ) )
@ ^ [X4: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X4 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_fst @ int @ int @ X4 ) ) ) ) ).
% inverse_rat.rsp
thf(fact_7319_Bseq__monoseq__convergent_H__inc,axiom,
! [F3: nat > real,M7: nat] :
( ( bfun @ nat @ real
@ ^ [N3: nat] : ( F3 @ ( plus_plus @ nat @ N3 @ M7 ) )
@ ( at_top @ nat ) )
=> ( ! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M7 @ M )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ real @ ( F3 @ M ) @ ( F3 @ N2 ) ) ) )
=> ( topolo6863149650580417670ergent @ real @ F3 ) ) ) ).
% Bseq_monoseq_convergent'_inc
thf(fact_7320_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
@ ^ [Y5: int,Z: int] : Y5 = Z
@ R
@ ( ring_1_of_int @ A )
@ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% transfer_rule_of_int
thf(fact_7321_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
@ ^ [Y5: num,Z: num] : Y5 = Z
@ R
@ ( numeral_numeral @ A )
@ ( numeral_numeral @ B ) ) ) ) ) ) ).
% transfer_rule_numeral
thf(fact_7322_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( power @ B )
& ( power @ A ) )
=> ! [R: A > B > $o] :
( ( 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 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y5: nat,Z: nat] : Y5 = Z
@ R )
@ ( power_power @ A )
@ ( power_power @ B ) ) ) ) ) ).
% power_transfer
thf(fact_7323_convergent__add__const__right__iff,axiom,
! [A: $tType] :
( ( topolo1287966508704411220up_add @ A )
=> ! [F3: nat > A,C3: A] :
( ( topolo6863149650580417670ergent @ A
@ ^ [N3: nat] : ( plus_plus @ A @ ( F3 @ N3 ) @ C3 ) )
= ( topolo6863149650580417670ergent @ A @ F3 ) ) ) ).
% convergent_add_const_right_iff
thf(fact_7324_convergent__add__const__iff,axiom,
! [A: $tType] :
( ( topolo1287966508704411220up_add @ A )
=> ! [C3: A,F3: nat > A] :
( ( topolo6863149650580417670ergent @ A
@ ^ [N3: nat] : ( plus_plus @ A @ C3 @ ( F3 @ N3 ) ) )
= ( topolo6863149650580417670ergent @ A @ F3 ) ) ) ).
% convergent_add_const_iff
thf(fact_7325_convergent__add,axiom,
! [A: $tType] :
( ( topolo6943815403480290642id_add @ A )
=> ! [X6: nat > A,Y8: nat > A] :
( ( topolo6863149650580417670ergent @ A @ X6 )
=> ( ( topolo6863149650580417670ergent @ A @ Y8 )
=> ( topolo6863149650580417670ergent @ A
@ ^ [N3: nat] : ( plus_plus @ A @ ( X6 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ) ).
% convergent_add
thf(fact_7326_convergent__ignore__initial__segment,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,M2: nat] :
( ( topolo6863149650580417670ergent @ A
@ ^ [N3: nat] : ( F3 @ ( plus_plus @ nat @ N3 @ M2 ) ) )
= ( topolo6863149650580417670ergent @ A @ F3 ) ) ) ).
% convergent_ignore_initial_segment
thf(fact_7327_convergent__Suc__iff,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A] :
( ( topolo6863149650580417670ergent @ A
@ ^ [N3: nat] : ( F3 @ ( suc @ N3 ) ) )
= ( topolo6863149650580417670ergent @ A @ F3 ) ) ) ).
% convergent_Suc_iff
thf(fact_7328_Bseq__mono__convergent,axiom,
! [X6: nat > real] :
( ( bfun @ nat @ real @ X6 @ ( at_top @ nat ) )
=> ( ! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ real @ ( X6 @ M ) @ ( X6 @ N2 ) ) )
=> ( topolo6863149650580417670ergent @ real @ X6 ) ) ) ).
% Bseq_mono_convergent
thf(fact_7329_convergent__realpow,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( topolo6863149650580417670ergent @ real @ ( power_power @ real @ X3 ) ) ) ) ).
% convergent_realpow
thf(fact_7330_uminus__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
@ ^ [X4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X4 ) ) @ ( product_snd @ int @ int @ X4 ) )
@ ^ [X4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X4 ) ) @ ( product_snd @ int @ int @ X4 ) ) ) ).
% uminus_rat.rsp
thf(fact_7331_Bseq__monoseq__convergent_H__dec,axiom,
! [F3: nat > real,M7: nat] :
( ( bfun @ nat @ real
@ ^ [N3: nat] : ( F3 @ ( plus_plus @ nat @ N3 @ M7 ) )
@ ( at_top @ nat ) )
=> ( ! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M7 @ M )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ real @ ( F3 @ N2 ) @ ( F3 @ M ) ) ) )
=> ( topolo6863149650580417670ergent @ real @ F3 ) ) ) ).
% Bseq_monoseq_convergent'_dec
thf(fact_7332_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
@ ^ [Y5: nat,Z: nat] : Y5 = Z
@ R
@ ( semiring_1_of_nat @ A )
@ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% transfer_rule_of_nat
thf(fact_7333_plus__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ratrel @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel )
@ ^ [X4: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y3 ) ) )
@ ^ [X4: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ).
% plus_rat.rsp
thf(fact_7334_less__eq__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y5: nat,Z: nat] : Y5 = Z
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y5: nat,Z: nat] : Y5 = Z
@ ^ [Y5: $o,Z: $o] : Y5 = Z )
@ ( ord_less_eq @ nat )
@ ( ord_less_eq @ nat ) ) ).
% less_eq_natural.rsp
thf(fact_7335_plus__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y5: nat,Z: nat] : Y5 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z: nat] : Y5 = Z
@ ^ [Y5: nat,Z: nat] : Y5 = Z )
@ ( plus_plus @ nat )
@ ( plus_plus @ nat ) ) ).
% plus_natural.rsp
thf(fact_7336_Suc_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z: nat] : Y5 = Z
@ ^ [Y5: nat,Z: nat] : Y5 = Z
@ suc
@ suc ) ).
% Suc.rsp
thf(fact_7337_sub_Orsp,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > int )
@ ^ [Y5: num,Z: num] : Y5 = Z
@ ( bNF_rel_fun @ num @ num @ int @ int
@ ^ [Y5: num,Z: num] : Y5 = Z
@ ^ [Y5: int,Z: int] : Y5 = Z )
@ ^ [M5: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N3 ) )
@ ^ [M5: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N3 ) ) ) ).
% sub.rsp
thf(fact_7338_Fract_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > ( product_prod @ int @ int ) )
@ ^ [Y5: int,Z: int] : Y5 = Z
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
@ ^ [Y5: int,Z: int] : Y5 = Z
@ ratrel )
@ ^ [A8: int,B8: int] :
( if @ ( product_prod @ int @ int )
@ ( B8
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A8 @ B8 ) )
@ ^ [A8: int,B8: int] :
( if @ ( product_prod @ int @ int )
@ ( B8
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A8 @ B8 ) ) ) ).
% Fract.rsp
thf(fact_7339_times__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ratrel @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel )
@ ^ [X4: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y3 ) ) )
@ ^ [X4: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ).
% times_rat.rsp
thf(fact_7340_plus__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ pcr_rat @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat )
@ ^ [X4: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y3 ) ) )
@ ( plus_plus @ rat ) ) ).
% plus_rat.transfer
thf(fact_7341_inverse__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X4: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X4 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_fst @ int @ int @ X4 ) ) )
@ ( inverse_inverse @ rat ) ) ).
% inverse_rat.transfer
thf(fact_7342_one__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( one_one @ rat ) ).
% one_rat.transfer
thf(fact_7343_zero__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( zero_zero @ rat ) ).
% zero_rat.transfer
thf(fact_7344_uminus__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X4 ) ) @ ( product_snd @ int @ int @ X4 ) )
@ ( uminus_uminus @ rat ) ) ).
% uminus_rat.transfer
thf(fact_7345_times__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ pcr_rat @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat )
@ ^ [X4: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y3 ) ) )
@ ( times_times @ rat ) ) ).
% times_rat.transfer
thf(fact_7346_times__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y3 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V5 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) )
@ ( times_times @ int ) ) ).
% times_int.transfer
thf(fact_7347_minus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) )
@ ( minus_minus @ int ) ) ).
% minus_int.transfer
thf(fact_7348_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% zero_int.transfer
thf(fact_7349_int__transfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
@ ^ [Y5: nat,Z: nat] : Y5 = Z
@ pcr_int
@ ^ [N3: nat] : ( product_Pair @ nat @ nat @ N3 @ ( zero_zero @ nat ) )
@ ( semiring_1_of_nat @ int ) ) ).
% int_transfer
thf(fact_7350_uminus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X4 ) )
@ ( uminus_uminus @ int ) ) ).
% uminus_int.transfer
thf(fact_7351_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% one_int.transfer
thf(fact_7352_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
@ ^ [Y5: $o,Z: $o] : Y5 = Z )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
@ ( ord_less @ int ) ) ).
% less_int.transfer
thf(fact_7353_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
@ ^ [Y5: $o,Z: $o] : Y5 = Z )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
@ ( ord_less_eq @ int ) ) ).
% less_eq_int.transfer
thf(fact_7354_plus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V5 ) ) ) )
@ ( plus_plus @ int ) ) ).
% plus_int.transfer
thf(fact_7355_times__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y3 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V5 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y3 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V5 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_7356_minus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_7357_intrel__iff,axiom,
! [X3: nat,Y: nat,U: nat,V2: nat] :
( ( intrel @ ( product_Pair @ nat @ nat @ X3 @ Y ) @ ( product_Pair @ nat @ nat @ U @ V2 ) )
= ( ( plus_plus @ nat @ X3 @ V2 )
= ( plus_plus @ nat @ U @ Y ) ) ) ).
% intrel_iff
thf(fact_7358_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_7359_uminus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X4 ) )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X4 ) ) ) ).
% uminus_int.rsp
thf(fact_7360_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_7361_intrel__def,axiom,
( intrel
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] :
( ( plus_plus @ nat @ X4 @ V5 )
= ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% intrel_def
thf(fact_7362_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
@ ^ [Y5: $o,Z: $o] : Y5 = Z )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% less_int.rsp
thf(fact_7363_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
@ ^ [Y5: $o,Z: $o] : Y5 = Z )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_7364_plus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V5 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V5 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_7365_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( comm_semiring_0 @ B )
& ( comm_semiring_0 @ A ) )
=> ! [A6: A > B > $o,B5: C > D > $o] :
( ( A6 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A6 @ ( bNF_rel_fun @ A @ B @ A @ B @ A6 @ A6 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A6 @ ( bNF_rel_fun @ A @ B @ A @ B @ A6 @ A6 ) @ ( 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 @ B5 @ A6 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A6 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B5 ) @ A6 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% horner_sum_transfer
thf(fact_7366_VEBT_Osimps_I7_J,axiom,
! [A: $tType,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ ( product_prod @ vEBT_VEBT @ A ) ) > vEBT_VEBT > A > A,F22: $o > $o > A,X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] :
( ( vEBT_rec_VEBT @ A @ F1 @ F22 @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( F1 @ X11 @ X12
@ ( map @ vEBT_VEBT @ ( product_prod @ vEBT_VEBT @ A )
@ ^ [VEBT: vEBT_VEBT] : ( product_Pair @ vEBT_VEBT @ A @ VEBT @ ( vEBT_rec_VEBT @ A @ F1 @ F22 @ VEBT ) )
@ X13 )
@ X14
@ ( vEBT_rec_VEBT @ A @ F1 @ F22 @ X14 ) ) ) ).
% VEBT.simps(7)
thf(fact_7367_length__transfer,axiom,
! [A: $tType,B: $tType,A6: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ nat @ nat @ ( list_all2 @ A @ B @ A6 )
@ ^ [Y5: nat,Z: nat] : Y5 = Z
@ ( size_size @ ( list @ A ) )
@ ( size_size @ ( list @ B ) ) ) ).
% length_transfer
thf(fact_7368_list__all2__same,axiom,
! [A: $tType,P: A > A > $o,Xs2: list @ A] :
( ( list_all2 @ A @ A @ P @ Xs2 @ Xs2 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X4 @ X4 ) ) ) ) ).
% list_all2_same
thf(fact_7369_list_Orel__refl__strong,axiom,
! [A: $tType,X3: list @ A,Ra2: A > A > $o] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ X3 ) )
=> ( Ra2 @ Z3 @ Z3 ) )
=> ( list_all2 @ A @ A @ Ra2 @ X3 @ X3 ) ) ).
% list.rel_refl_strong
thf(fact_7370_list_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X3: list @ A,Y: list @ B,Ra2: A > B > $o] :
( ( list_all2 @ A @ B @ R @ X3 @ Y )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( set2 @ A @ X3 ) )
=> ( ( member @ B @ Yb @ ( set2 @ B @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra2 @ Z3 @ Yb ) ) ) )
=> ( list_all2 @ A @ B @ Ra2 @ X3 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_7371_list_Orel__cong,axiom,
! [A: $tType,B: $tType,X3: list @ A,Ya: list @ A,Y: list @ B,Xa2: list @ B,R: A > B > $o,Ra2: A > B > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( set2 @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( set2 @ B @ Xa2 ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra2 @ Z3 @ Yb ) ) ) )
=> ( ( list_all2 @ A @ B @ R @ X3 @ Y )
= ( list_all2 @ A @ B @ Ra2 @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_7372_list__all2__append,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,P: A > B > $o,Us: list @ A,Vs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( list_all2 @ A @ B @ P @ ( append @ A @ Xs2 @ Us ) @ ( append @ B @ Ys @ Vs ) )
= ( ( list_all2 @ A @ B @ P @ Xs2 @ Ys )
& ( list_all2 @ A @ B @ P @ Us @ Vs ) ) ) ) ).
% list_all2_append
thf(fact_7373_list__all2__append1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs2: list @ A,Ys: list @ A,Zs2: list @ B] :
( ( list_all2 @ A @ B @ P @ ( append @ A @ Xs2 @ Ys ) @ Zs2 )
= ( ? [Us2: list @ B,Vs2: list @ B] :
( ( Zs2
= ( append @ B @ Us2 @ Vs2 ) )
& ( ( size_size @ ( list @ B ) @ Us2 )
= ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ( size_size @ ( list @ B ) @ Vs2 )
= ( size_size @ ( list @ A ) @ Ys ) )
& ( list_all2 @ A @ B @ P @ Xs2 @ Us2 )
& ( list_all2 @ A @ B @ P @ Ys @ Vs2 ) ) ) ) ).
% list_all2_append1
thf(fact_7374_list__all2__append2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs2: list @ A,Ys: list @ B,Zs2: list @ B] :
( ( list_all2 @ A @ B @ P @ Xs2 @ ( append @ B @ Ys @ Zs2 ) )
= ( ? [Us2: list @ A,Vs2: list @ A] :
( ( Xs2
= ( append @ A @ Us2 @ Vs2 ) )
& ( ( size_size @ ( list @ A ) @ Us2 )
= ( size_size @ ( list @ B ) @ Ys ) )
& ( ( size_size @ ( list @ A ) @ Vs2 )
= ( size_size @ ( list @ B ) @ Zs2 ) )
& ( list_all2 @ A @ B @ P @ Us2 @ Ys )
& ( list_all2 @ A @ B @ P @ Vs2 @ Zs2 ) ) ) ) ).
% list_all2_append2
thf(fact_7375_list__all2__lengthD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs2: list @ A,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ Xs2 @ Ys )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_7376_VEBT_Osimps_I8_J,axiom,
! [A: $tType,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ ( product_prod @ vEBT_VEBT @ A ) ) > vEBT_VEBT > A > A,F22: $o > $o > A,X21: $o,X222: $o] :
( ( vEBT_rec_VEBT @ A @ F1 @ F22 @ ( vEBT_Leaf @ X21 @ X222 ) )
= ( F22 @ X21 @ X222 ) ) ).
% VEBT.simps(8)
thf(fact_7377_list__all2__conv__all__nth,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P4: A > B > $o,Xs: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P4 @ ( nth @ A @ Xs @ I4 ) @ ( nth @ B @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_7378_list__all2__all__nthI,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P: A > B > $o] :
( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ A3 ) )
=> ( P @ ( nth @ A @ A3 @ N2 ) @ ( nth @ B @ B2 @ N2 ) ) )
=> ( list_all2 @ A @ B @ P @ A3 @ B2 ) ) ) ).
% list_all2_all_nthI
thf(fact_7379_list__all2__nthD2,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs2: list @ A,Ys: list @ B,P2: nat] :
( ( list_all2 @ A @ B @ P @ Xs2 @ Ys )
=> ( ( ord_less @ nat @ P2 @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( P @ ( nth @ A @ Xs2 @ P2 ) @ ( nth @ B @ Ys @ P2 ) ) ) ) ).
% list_all2_nthD2
thf(fact_7380_list__all2__nthD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs2: list @ A,Ys: list @ B,P2: nat] :
( ( list_all2 @ A @ B @ P @ Xs2 @ Ys )
=> ( ( ord_less @ nat @ P2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ P2 ) @ ( nth @ B @ Ys @ P2 ) ) ) ) ).
% list_all2_nthD
thf(fact_7381_product__lists__set,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss ) )
= ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( list_all2 @ A @ ( list @ A )
@ ^ [X4: A,Ys3: list @ A] : ( member @ A @ X4 @ ( set2 @ A @ Ys3 ) )
@ Xs
@ Xss ) ) ) ).
% product_lists_set
thf(fact_7382_list__all2I,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P: A > B > $o] :
( ! [X5: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X5 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ A3 @ B2 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P @ X5 ) )
=> ( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( list_all2 @ A @ B @ P @ A3 @ B2 ) ) ) ).
% list_all2I
thf(fact_7383_sum__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( monoid_add @ A ) )
=> ! [A6: A > B > $o] :
( ( A6 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A6 @ ( bNF_rel_fun @ A @ B @ A @ B @ A6 @ A6 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A6 ) @ A6 @ ( groups8242544230860333062m_list @ A ) @ ( groups8242544230860333062m_list @ B ) ) ) ) ) ).
% sum_list_transfer
thf(fact_7384_list__all2__iff,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P4: A > B > $o,Xs: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [X4: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X4 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys3 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P4 @ X4 ) ) ) ) ) ).
% list_all2_iff
thf(fact_7385_admissible__chfin,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o] :
( ! [S4: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ S4 )
=> ( finite_finite2 @ A @ S4 ) )
=> ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P ) ) ) ).
% admissible_chfin
thf(fact_7386_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X3: A > B,Xa2: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X3 @ Xa2 )
= Y )
=> ( ! [X5: A] :
( ( Xa2
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( Y != X5 ) )
=> ( ! [X5: A,Y4: A,Zs: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Zs ) ) )
=> ( Y
!= ( if @ A @ ( ord_less_eq @ B @ ( X3 @ X5 ) @ ( X3 @ ( arg_min_list @ A @ B @ X3 @ ( cons @ A @ Y4 @ Zs ) ) ) ) @ X5 @ ( arg_min_list @ A @ B @ X3 @ ( cons @ A @ Y4 @ Zs ) ) ) ) )
=> ~ ( ( Xa2
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_7387_ccpo_OadmissibleD,axiom,
! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o,P: A > $o,A6: set @ A] :
( ( comple1908693960933563346ssible @ A @ Lub @ Ord @ P )
=> ( ( comple1602240252501008431_chain @ A @ Ord @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( P @ X5 ) )
=> ( P @ ( Lub @ A6 ) ) ) ) ) ) ).
% ccpo.admissibleD
thf(fact_7388_ccpo_OadmissibleI,axiom,
! [A: $tType,Ord: A > A > $o,P: A > $o,Lub: ( set @ A ) > A] :
( ! [A10: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord @ A10 )
=> ( ( A10
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ A10 )
=> ( P @ X ) )
=> ( P @ ( Lub @ A10 ) ) ) ) )
=> ( comple1908693960933563346ssible @ A @ Lub @ Ord @ P ) ) ).
% ccpo.admissibleI
thf(fact_7389_ccpo_Oadmissible__def,axiom,
! [A: $tType] :
( ( comple1908693960933563346ssible @ A )
= ( ^ [Lub2: ( set @ A ) > A,Ord2: A > A > $o,P4: A > $o] :
! [A7: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord2 @ A7 )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( P4 @ X4 ) )
=> ( P4 @ ( Lub2 @ A7 ) ) ) ) ) ) ) ).
% ccpo.admissible_def
thf(fact_7390_option_Othe__def,axiom,
! [A: $tType] :
( ( the2 @ A )
= ( case_option @ A @ A @ ( undefined @ A )
@ ^ [X23: A] : X23 ) ) ).
% option.the_def
thf(fact_7391_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 )
@ ^ [X4: A] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) ) ) ) ).
% admissible_disj
thf(fact_7392_Func__empty,axiom,
! [B: $tType,A: $tType,B5: set @ B] :
( ( bNF_Wellorder_Func @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B5 )
= ( insert2 @ ( A > B )
@ ^ [X4: A] : ( undefined @ B )
@ ( bot_bot @ ( set @ ( A > B ) ) ) ) ) ).
% Func_empty
thf(fact_7393_pair__lessI2,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( ord_less @ nat @ S @ T2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_7394_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X4: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X4 )
@ ^ [X4: A,Y3: A] : ( ord_less @ A @ Y3 @ X4 )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_7395_pair__less__iff1,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X3 @ Y ) @ ( product_Pair @ nat @ nat @ X3 @ Z2 ) ) @ fun_pair_less )
= ( ord_less @ nat @ Y @ Z2 ) ) ).
% pair_less_iff1
thf(fact_7396_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ( ( Less_eq2 @ Top @ A3 )
=> ( A3 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_7397_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ( ( A3 != Top )
= ( Less @ A3 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_7398_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ( ( Less_eq2 @ Top @ A3 )
= ( A3 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_7399_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ~ ( Less @ Top @ A3 ) ) ).
% ordering_top.extremum_strict
thf(fact_7400_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ( Less_eq2 @ A3 @ Top ) ) ).
% ordering_top.extremum
thf(fact_7401_pair__lessI1,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_7402_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_7403_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top @ nat
@ ^ [X4: nat,Y3: nat] : ( ord_less_eq @ nat @ Y3 @ X4 )
@ ^ [X4: nat,Y3: nat] : ( ord_less @ nat @ Y3 @ X4 )
@ ( zero_zero @ nat ) ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_7404_pair__leqI2,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( ord_less_eq @ nat @ S @ T2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_7405_pair__leqI1,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_7406_wmin__insertI,axiom,
! [X3: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X3 @ XS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_leq )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_min_weak )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ ( insert2 @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).
% wmin_insertI
thf(fact_7407_wmax__insertI,axiom,
! [Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X3: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y @ YS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_leq )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_max_weak )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert2 @ ( product_prod @ nat @ nat ) @ X3 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% wmax_insertI
thf(fact_7408_wmax__emptyI,axiom,
! [X6: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ X6 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ X6 ) @ fun_max_weak ) ) ).
% wmax_emptyI
thf(fact_7409_wmin__emptyI,axiom,
! [X6: set @ ( product_prod @ nat @ nat )] : ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X6 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ).
% wmin_emptyI
thf(fact_7410_min__weak__def,axiom,
( fun_min_weak
= ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert2 @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% min_weak_def
thf(fact_7411_max__weak__def,axiom,
( fun_max_weak
= ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert2 @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% max_weak_def
thf(fact_7412_smin__insertI,axiom,
! [X3: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X3 @ XS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_less )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_min_strict )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ ( insert2 @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).
% smin_insertI
thf(fact_7413_smax__insertI,axiom,
! [Y: product_prod @ nat @ nat,Y8: set @ ( product_prod @ nat @ nat ),X3: product_prod @ nat @ nat,X6: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y @ Y8 )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_less )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X6 @ Y8 ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert2 @ ( product_prod @ nat @ nat ) @ X3 @ X6 ) @ Y8 ) @ fun_max_strict ) ) ) ) ).
% smax_insertI
thf(fact_7414_smin__emptyI,axiom,
! [X6: set @ ( product_prod @ nat @ nat )] :
( ( X6
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X6 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ).
% smin_emptyI
thf(fact_7415_smax__emptyI,axiom,
! [Y8: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ Y8 )
=> ( ( Y8
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ Y8 ) @ fun_max_strict ) ) ) ).
% smax_emptyI
thf(fact_7416_max__rpair__set,axiom,
fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_max_strict @ fun_max_weak ) ).
% max_rpair_set
thf(fact_7417_min__rpair__set,axiom,
fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict @ fun_min_weak ) ).
% min_rpair_set
thf(fact_7418_of__rat__neg__numeral__eq,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [W: num] :
( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ W ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ).
% of_rat_neg_numeral_eq
thf(fact_7419_span__explicit,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_span @ A )
= ( ^ [B8: set @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [T3: set @ A,R5: A > real] :
( ( Uu3
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [A8: A] : ( real_V8093663219630862766scaleR @ A @ ( R5 @ A8 ) @ A8 )
@ T3 ) )
& ( finite_finite2 @ A @ T3 )
& ( ord_less_eq @ ( set @ A ) @ T3 @ B8 ) ) ) ) ) ) ).
% span_explicit
thf(fact_7420_span__insert__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] :
( ( real_Vector_span @ A @ ( insert2 @ A @ ( zero_zero @ A ) @ S3 ) )
= ( real_Vector_span @ A @ S3 ) ) ) ).
% span_insert_0
thf(fact_7421_of__rat__numeral__eq,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [W: num] :
( ( field_char_0_of_rat @ A @ ( numeral_numeral @ rat @ W ) )
= ( numeral_numeral @ A @ W ) ) ) ).
% of_rat_numeral_eq
thf(fact_7422_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_7423_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_7424_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_7425_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_7426_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_7427_span__delete__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] :
( ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( real_Vector_span @ A @ S3 ) ) ) ).
% span_delete_0
thf(fact_7428_in__span__delete,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A,S3: set @ A,B2: A] :
( ( member @ A @ A3 @ ( real_Vector_span @ A @ S3 ) )
=> ( ~ ( member @ A @ A3 @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ A @ B2 @ ( real_Vector_span @ A @ ( insert2 @ A @ A3 @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% in_span_delete
thf(fact_7429_span__redundant,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A,S3: set @ A] :
( ( member @ A @ X3 @ ( real_Vector_span @ A @ S3 ) )
=> ( ( real_Vector_span @ A @ ( insert2 @ A @ X3 @ S3 ) )
= ( real_Vector_span @ A @ S3 ) ) ) ) ).
% span_redundant
thf(fact_7430_in__span__insert,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A,B2: A,S3: set @ A] :
( ( member @ A @ A3 @ ( real_Vector_span @ A @ ( insert2 @ A @ B2 @ S3 ) ) )
=> ( ~ ( member @ A @ A3 @ ( real_Vector_span @ A @ S3 ) )
=> ( member @ A @ B2 @ ( real_Vector_span @ A @ ( insert2 @ A @ A3 @ S3 ) ) ) ) ) ) ).
% in_span_insert
thf(fact_7431_span__trans,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A,S3: set @ A,Y: A] :
( ( member @ A @ X3 @ ( real_Vector_span @ A @ S3 ) )
=> ( ( member @ A @ Y @ ( real_Vector_span @ A @ ( insert2 @ A @ X3 @ S3 ) ) )
=> ( member @ A @ Y @ ( real_Vector_span @ A @ S3 ) ) ) ) ) ).
% span_trans
thf(fact_7432_eq__span__insert__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A,Y: A,S3: set @ A] :
( ( member @ A @ ( minus_minus @ A @ X3 @ Y ) @ ( real_Vector_span @ A @ S3 ) )
=> ( ( real_Vector_span @ A @ ( insert2 @ A @ X3 @ S3 ) )
= ( real_Vector_span @ A @ ( insert2 @ A @ Y @ S3 ) ) ) ) ) ).
% eq_span_insert_eq
thf(fact_7433_independent__insertI,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A,S3: set @ A] :
( ~ ( member @ A @ A3 @ ( real_Vector_span @ A @ S3 ) )
=> ( ~ ( real_V358717886546972837endent @ A @ S3 )
=> ~ ( real_V358717886546972837endent @ A @ ( insert2 @ A @ A3 @ S3 ) ) ) ) ) ).
% independent_insertI
thf(fact_7434_independent__insert,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A,S3: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ ( insert2 @ A @ A3 @ S3 ) ) )
= ( ( ( member @ A @ A3 @ S3 )
=> ~ ( real_V358717886546972837endent @ A @ S3 ) )
& ( ~ ( member @ A @ A3 @ S3 )
=> ( ~ ( real_V358717886546972837endent @ A @ S3 )
& ~ ( member @ A @ A3 @ ( real_Vector_span @ A @ S3 ) ) ) ) ) ) ) ).
% independent_insert
thf(fact_7435_dependent__insertD,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A,S3: set @ A] :
( ~ ( member @ A @ A3 @ ( real_Vector_span @ A @ S3 ) )
=> ( ( real_V358717886546972837endent @ A @ ( insert2 @ A @ A3 @ S3 ) )
=> ( real_V358717886546972837endent @ A @ S3 ) ) ) ) ).
% dependent_insertD
thf(fact_7436_span__breakdown__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A,A3: A,S3: set @ A] :
( ( member @ A @ X3 @ ( real_Vector_span @ A @ ( insert2 @ A @ A3 @ S3 ) ) )
= ( ? [K3: real] : ( member @ A @ ( minus_minus @ A @ X3 @ ( real_V8093663219630862766scaleR @ A @ K3 @ A3 ) ) @ ( real_Vector_span @ A @ S3 ) ) ) ) ) ).
% span_breakdown_eq
thf(fact_7437_of__rat__add,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat,B2: rat] :
( ( field_char_0_of_rat @ A @ ( plus_plus @ rat @ A3 @ B2 ) )
= ( plus_plus @ A @ ( field_char_0_of_rat @ A @ A3 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ).
% of_rat_add
thf(fact_7438_of__rat__power,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat,N: nat] :
( ( field_char_0_of_rat @ A @ ( power_power @ rat @ A3 @ N ) )
= ( power_power @ A @ ( field_char_0_of_rat @ A @ A3 ) @ N ) ) ) ).
% of_rat_power
thf(fact_7439_span__add__eq2,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Y: A,S3: set @ A,X3: A] :
( ( member @ A @ Y @ ( real_Vector_span @ A @ S3 ) )
=> ( ( member @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( real_Vector_span @ A @ S3 ) )
= ( member @ A @ X3 @ ( real_Vector_span @ A @ S3 ) ) ) ) ) ).
% span_add_eq2
thf(fact_7440_span__add__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A,S3: set @ A,Y: A] :
( ( member @ A @ X3 @ ( real_Vector_span @ A @ S3 ) )
=> ( ( member @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( real_Vector_span @ A @ S3 ) )
= ( member @ A @ Y @ ( real_Vector_span @ A @ S3 ) ) ) ) ) ).
% span_add_eq
thf(fact_7441_span__add,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A,S3: set @ A,Y: A] :
( ( member @ A @ X3 @ ( real_Vector_span @ A @ S3 ) )
=> ( ( member @ A @ Y @ ( real_Vector_span @ A @ S3 ) )
=> ( member @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( real_Vector_span @ A @ S3 ) ) ) ) ) ).
% span_add
thf(fact_7442_maximal__independent__subset,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [V3: set @ A] :
~ ! [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ V3 )
=> ( ~ ( real_V358717886546972837endent @ A @ B7 )
=> ~ ( ord_less_eq @ ( set @ A ) @ V3 @ ( real_Vector_span @ A @ B7 ) ) ) ) ) ).
% maximal_independent_subset
thf(fact_7443_spanning__subset__independent,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B5: set @ A,A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ~ ( real_V358717886546972837endent @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( real_Vector_span @ A @ B5 ) )
=> ( A6 = B5 ) ) ) ) ) ).
% spanning_subset_independent
thf(fact_7444_maximal__independent__subset__extend,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A,V3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ V3 )
=> ( ~ ( real_V358717886546972837endent @ A @ S3 )
=> ~ ! [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ V3 )
=> ( ~ ( real_V358717886546972837endent @ A @ B7 )
=> ~ ( ord_less_eq @ ( set @ A ) @ V3 @ ( real_Vector_span @ A @ B7 ) ) ) ) ) ) ) ) ).
% maximal_independent_subset_extend
thf(fact_7445_of__rat__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat,S: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( field_char_0_of_rat @ A @ S ) )
= ( ord_less_eq @ rat @ R2 @ S ) ) ) ).
% of_rat_less_eq
thf(fact_7446_span__superset,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] : ( ord_less_eq @ ( set @ A ) @ S3 @ ( real_Vector_span @ A @ S3 ) ) ) ).
% span_superset
thf(fact_7447_span__mono,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( real_Vector_span @ A @ A6 ) @ ( real_Vector_span @ A @ B5 ) ) ) ) ).
% span_mono
thf(fact_7448_span__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A,T4: set @ A] :
( ( ( real_Vector_span @ A @ S3 )
= ( real_Vector_span @ A @ T4 ) )
= ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( real_Vector_span @ A @ T4 ) )
& ( ord_less_eq @ ( set @ A ) @ T4 @ ( real_Vector_span @ A @ S3 ) ) ) ) ) ).
% span_eq
thf(fact_7449_span__induct__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A,S3: set @ A,H: A > $o] :
( ( member @ A @ X3 @ ( real_Vector_span @ A @ S3 ) )
=> ( ( H @ ( zero_zero @ A ) )
=> ( ! [C2: real,X5: A,Y4: A] :
( ( member @ A @ X5 @ S3 )
=> ( ( H @ Y4 )
=> ( H @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ X5 ) @ Y4 ) ) ) )
=> ( H @ X3 ) ) ) ) ) ).
% span_induct_alt
thf(fact_7450_span__Un,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A,T4: set @ A] :
( ( real_Vector_span @ A @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [X4: A,Y3: A] :
( ( Uu3
= ( plus_plus @ A @ X4 @ Y3 ) )
& ( member @ A @ X4 @ ( real_Vector_span @ A @ S3 ) )
& ( member @ A @ Y3 @ ( real_Vector_span @ A @ T4 ) ) ) ) ) ) ).
% span_Un
thf(fact_7451_span__insert,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A,S3: set @ A] :
( ( real_Vector_span @ A @ ( insert2 @ A @ A3 @ S3 ) )
= ( collect @ A
@ ^ [X4: A] :
? [K3: real] : ( member @ A @ ( minus_minus @ A @ X4 @ ( real_V8093663219630862766scaleR @ A @ K3 @ A3 ) ) @ ( real_Vector_span @ A @ S3 ) ) ) ) ) ).
% span_insert
thf(fact_7452_dependent__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V358717886546972837endent @ A )
= ( ^ [P4: set @ A] :
? [X4: A] :
( ( member @ A @ X4 @ P4 )
& ( member @ A @ X4 @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ P4 @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% dependent_def
thf(fact_7453_span__singleton,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A] :
( ( real_Vector_span @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image2 @ real @ A
@ ^ [K3: real] : ( real_V8093663219630862766scaleR @ A @ K3 @ X3 )
@ ( top_top @ ( set @ real ) ) ) ) ) ).
% span_singleton
thf(fact_7454_span__breakdown,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: A,S3: set @ A,A3: A] :
( ( member @ A @ B2 @ S3 )
=> ( ( member @ A @ A3 @ ( real_Vector_span @ A @ S3 ) )
=> ? [K: real] : ( member @ A @ ( minus_minus @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ K @ B2 ) ) @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% span_breakdown
thf(fact_7455_independent__span__bound,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [T4: set @ A,S3: set @ A] :
( ( finite_finite2 @ A @ T4 )
=> ( ~ ( real_V358717886546972837endent @ A @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( real_Vector_span @ A @ T4 ) )
=> ( ( finite_finite2 @ A @ S3 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ S3 ) @ ( finite_card @ A @ T4 ) ) ) ) ) ) ) ).
% independent_span_bound
thf(fact_7456_exchange__lemma,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [T4: set @ A,S3: set @ A] :
( ( finite_finite2 @ A @ T4 )
=> ( ~ ( real_V358717886546972837endent @ A @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( real_Vector_span @ A @ T4 ) )
=> ? [T12: set @ A] :
( ( ( finite_card @ A @ T12 )
= ( finite_card @ A @ T4 ) )
& ( finite_finite2 @ A @ T12 )
& ( ord_less_eq @ ( set @ A ) @ S3 @ T12 )
& ( ord_less_eq @ ( set @ A ) @ T12 @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) )
& ( ord_less_eq @ ( set @ A ) @ S3 @ ( real_Vector_span @ A @ T12 ) ) ) ) ) ) ) ).
% exchange_lemma
thf(fact_7457_span__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_span @ A )
= ( ^ [B6: set @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [F4: A > real] :
( ( Uu3
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( F4 @ X4 ) @ X4 )
@ ( collect @ A
@ ^ [X4: A] :
( ( F4 @ X4 )
!= ( zero_zero @ real ) ) ) ) )
& ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( F4 @ X4 )
!= ( zero_zero @ real ) ) )
@ B6 )
& ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( F4 @ X4 )
!= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% span_alt
thf(fact_7458_sum__representation__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A,B5: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ Basis ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ Basis @ B5 )
=> ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [B8: A] : ( real_V8093663219630862766scaleR @ A @ ( real_V7696804695334737415tation @ A @ Basis @ V2 @ B8 ) @ B8 )
@ B5 )
= V2 ) ) ) ) ) ) ).
% sum_representation_eq
thf(fact_7459_dim__le__card,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [V3: set @ A,W4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ V3 @ ( real_Vector_span @ A @ W4 ) )
=> ( ( finite_finite2 @ A @ W4 )
=> ( ord_less_eq @ nat @ ( real_Vector_dim @ A @ V3 ) @ ( finite_card @ A @ W4 ) ) ) ) ) ).
% dim_le_card
thf(fact_7460_representation__extend,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A,Basis2: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ Basis2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ Basis2 @ Basis )
=> ( ( real_V7696804695334737415tation @ A @ Basis @ V2 )
= ( real_V7696804695334737415tation @ A @ Basis2 @ V2 ) ) ) ) ) ) ).
% representation_extend
thf(fact_7461_dim__le__card_H,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ord_less_eq @ nat @ ( real_Vector_dim @ A @ S ) @ ( finite_card @ A @ S ) ) ) ) ).
% dim_le_card'
thf(fact_7462_representation__add,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A,U: A] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ Basis ) )
=> ( ( member @ A @ U @ ( real_Vector_span @ A @ Basis ) )
=> ( ( real_V7696804695334737415tation @ A @ Basis @ ( plus_plus @ A @ U @ V2 ) )
= ( ^ [B8: A] : ( plus_plus @ real @ ( real_V7696804695334737415tation @ A @ Basis @ U @ B8 ) @ ( real_V7696804695334737415tation @ A @ Basis @ V2 @ B8 ) ) ) ) ) ) ) ) ).
% representation_add
thf(fact_7463_basis__card__eq__dim,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B5: set @ A,V3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ V3 )
=> ( ( ord_less_eq @ ( set @ A ) @ V3 @ ( real_Vector_span @ A @ B5 ) )
=> ( ~ ( real_V358717886546972837endent @ A @ B5 )
=> ( ( finite_card @ A @ B5 )
= ( real_Vector_dim @ A @ V3 ) ) ) ) ) ) ).
% basis_card_eq_dim
thf(fact_7464_basis__exists,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [V3: set @ A] :
~ ! [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ V3 )
=> ( ~ ( real_V358717886546972837endent @ A @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ V3 @ ( real_Vector_span @ A @ B7 ) )
=> ( ( finite_card @ A @ B7 )
!= ( real_Vector_dim @ A @ V3 ) ) ) ) ) ) ).
% basis_exists
thf(fact_7465_dim__unique,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B5: set @ A,V3: set @ A,N: nat] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ V3 )
=> ( ( ord_less_eq @ ( set @ A ) @ V3 @ ( real_Vector_span @ A @ B5 ) )
=> ( ~ ( real_V358717886546972837endent @ A @ B5 )
=> ( ( ( finite_card @ A @ B5 )
= N )
=> ( ( real_Vector_dim @ A @ V3 )
= N ) ) ) ) ) ) ).
% dim_unique
thf(fact_7466_span__card__ge__dim,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B5: set @ A,V3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ V3 )
=> ( ( ord_less_eq @ ( set @ A ) @ V3 @ ( real_Vector_span @ A @ B5 ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ nat @ ( real_Vector_dim @ A @ V3 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ) ).
% span_card_ge_dim
thf(fact_7467_AboveS__def,axiom,
! [A: $tType] :
( ( order_AboveS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A7: set @ A] :
( collect @ A
@ ^ [B8: A] :
( ( member @ A @ B8 @ ( field2 @ A @ R5 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ( B8 != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ B8 ) @ R5 ) ) ) ) ) ) ) ).
% AboveS_def
thf(fact_7468_rat__number__expand_I5_J,axiom,
! [K2: num] :
( ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K2 ) )
= ( fract @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) @ ( one_one @ int ) ) ) ).
% rat_number_expand(5)
thf(fact_7469_normalize__eq,axiom,
! [A3: int,B2: int,P2: int,Q3: int] :
( ( ( normalize @ ( product_Pair @ int @ int @ A3 @ B2 ) )
= ( product_Pair @ int @ int @ P2 @ Q3 ) )
=> ( ( fract @ P2 @ Q3 )
= ( fract @ A3 @ B2 ) ) ) ).
% normalize_eq
thf(fact_7470_quotient__of__eq,axiom,
! [A3: int,B2: int,P2: int,Q3: int] :
( ( ( quotient_of @ ( fract @ A3 @ B2 ) )
= ( product_Pair @ int @ int @ P2 @ Q3 ) )
=> ( ( fract @ P2 @ Q3 )
= ( fract @ A3 @ B2 ) ) ) ).
% quotient_of_eq
thf(fact_7471_rat__number__expand_I3_J,axiom,
( ( numeral_numeral @ rat )
= ( ^ [K3: num] : ( fract @ ( numeral_numeral @ int @ K3 ) @ ( one_one @ int ) ) ) ) ).
% rat_number_expand(3)
thf(fact_7472_rat__number__collapse_I3_J,axiom,
! [W: num] :
( ( fract @ ( numeral_numeral @ int @ W ) @ ( one_one @ int ) )
= ( numeral_numeral @ rat @ W ) ) ).
% rat_number_collapse(3)
thf(fact_7473_AboveS__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( order_AboveS @ A @ R2 @ A6 ) @ ( field2 @ A @ R2 ) ) ).
% AboveS_Field
thf(fact_7474_AboveS__disjoint,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( inf_inf @ ( set @ A ) @ A6 @ ( order_AboveS @ A @ R2 @ A6 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% AboveS_disjoint
thf(fact_7475_quotient__of__Fract,axiom,
! [A3: int,B2: int] :
( ( quotient_of @ ( fract @ A3 @ B2 ) )
= ( normalize @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ).
% quotient_of_Fract
thf(fact_7476_Fract_Oabs__eq,axiom,
( fract
= ( ^ [Xa4: int,X4: int] :
( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( X4
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ Xa4 @ X4 ) ) ) ) ) ).
% Fract.abs_eq
thf(fact_7477_Fract_Otransfer,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat )
@ ^ [Y5: int,Z: int] : Y5 = Z
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ rat
@ ^ [Y5: int,Z: int] : Y5 = Z
@ pcr_rat )
@ ^ [A8: int,B8: int] :
( if @ ( product_prod @ int @ int )
@ ( B8
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A8 @ B8 ) )
@ fract ) ).
% Fract.transfer
thf(fact_7478_rat__number__collapse_I4_J,axiom,
! [W: num] :
( ( fract @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ W ) ) ) ).
% rat_number_collapse(4)
thf(fact_7479_wo__rel_Osuc__greater,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( ( order_AboveS @ A @ R2 @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ B2 @ B5 )
=> ( ( ( bNF_Wellorder_wo_suc @ A @ R2 @ B5 )
!= B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R2 @ B5 ) ) @ R2 ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_7480_wo__rel_Osuc__inField,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( ( order_AboveS @ A @ R2 @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_Wellorder_wo_suc @ A @ R2 @ B5 ) @ ( field2 @ A @ R2 ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_7481_wo__rel_Osuc__least__AboveS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A3 @ ( order_AboveS @ A @ R2 @ B5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_Wellorder_wo_suc @ A @ R2 @ B5 ) @ A3 ) @ R2 ) ) ) ).
% wo_rel.suc_least_AboveS
thf(fact_7482_wo__rel_Oequals__suc__AboveS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ A3 @ ( order_AboveS @ A @ R2 @ B5 ) )
=> ( ! [A21: A] :
( ( member @ A @ A21 @ ( order_AboveS @ A @ R2 @ B5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A21 ) @ R2 ) )
=> ( A3
= ( bNF_Wellorder_wo_suc @ A @ R2 @ B5 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_7483_wo__rel_Osuc__AboveS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( ( order_AboveS @ A @ R2 @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_Wellorder_wo_suc @ A @ R2 @ B5 ) @ ( order_AboveS @ A @ R2 @ B5 ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_7484_wo__rel_Osuc__ofilter__in,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A6 )
=> ( ( ( order_AboveS @ A @ R2 @ A6 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R2 @ A6 ) ) @ R2 )
=> ( ( B2
!= ( bNF_Wellorder_wo_suc @ A @ R2 @ A6 ) )
=> ( member @ A @ B2 @ A6 ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_7485_less__eq__enat__def,axiom,
( ( ord_less_eq @ extended_enat )
= ( ^ [M5: extended_enat] :
( extended_case_enat @ $o
@ ^ [N1: nat] :
( extended_case_enat @ $o
@ ^ [M12: nat] : ( ord_less_eq @ nat @ M12 @ N1 )
@ $false
@ M5 )
@ $true ) ) ) ).
% less_eq_enat_def
thf(fact_7486_wo__rel_Oofilter__linord,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A,B5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A6 )
=> ( ( order_ofilter @ A @ R2 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
| ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ) ).
% wo_rel.ofilter_linord
thf(fact_7487_ofilter__def,axiom,
! [A: $tType] :
( ( order_ofilter @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R5 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R5 @ X4 ) @ A7 ) ) ) ) ) ).
% ofilter_def
thf(fact_7488_wo__rel_Oofilter__def,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A6 )
= ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( field2 @ A @ R2 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R2 @ X4 ) @ A6 ) ) ) ) ) ).
% wo_rel.ofilter_def
thf(fact_7489_sorted__wrt__iff__nth__Suc__transp,axiom,
! [A: $tType,P: A > A > $o,Xs2: list @ A] :
( ( transp @ A @ P )
=> ( ( sorted_wrt @ A @ P @ Xs2 )
= ( ! [I4: nat] :
( ( ord_less @ nat @ ( suc @ I4 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ I4 ) @ ( nth @ A @ Xs2 @ ( suc @ I4 ) ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_Suc_transp
thf(fact_7490_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_7491_in__inv__image,axiom,
! [A: $tType,B: $tType,X3: A,Y: A,R2: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( inv_image @ B @ A @ R2 @ F3 ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_7492_transp__inf,axiom,
! [A: $tType,R2: A > A > $o,S: A > A > $o] :
( ( transp @ A @ R2 )
=> ( ( transp @ A @ S )
=> ( transp @ A @ ( inf_inf @ ( A > A > $o ) @ R2 @ S ) ) ) ) ).
% transp_inf
thf(fact_7493_transpD,axiom,
! [A: $tType,R2: A > A > $o,X3: A,Y: A,Z2: A] :
( ( transp @ A @ R2 )
=> ( ( R2 @ X3 @ Y )
=> ( ( R2 @ Y @ Z2 )
=> ( R2 @ X3 @ Z2 ) ) ) ) ).
% transpD
thf(fact_7494_transpE,axiom,
! [A: $tType,R2: A > A > $o,X3: A,Y: A,Z2: A] :
( ( transp @ A @ R2 )
=> ( ( R2 @ X3 @ Y )
=> ( ( R2 @ Y @ Z2 )
=> ( R2 @ X3 @ Z2 ) ) ) ) ).
% transpE
thf(fact_7495_transpI,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [X5: A,Y4: A,Z3: A] :
( ( R2 @ X5 @ Y4 )
=> ( ( R2 @ Y4 @ Z3 )
=> ( R2 @ X5 @ Z3 ) ) )
=> ( transp @ A @ R2 ) ) ).
% transpI
thf(fact_7496_transp__def,axiom,
! [A: $tType] :
( ( transp @ A )
= ( ^ [R5: A > A > $o] :
! [X4: A,Y3: A,Z4: A] :
( ( R5 @ X4 @ Y3 )
=> ( ( R5 @ Y3 @ Z4 )
=> ( R5 @ X4 @ Z4 ) ) ) ) ) ).
% transp_def
thf(fact_7497_transp__equality,axiom,
! [A: $tType] :
( transp @ A
@ ^ [Y5: A,Z: A] : Y5 = Z ) ).
% transp_equality
thf(fact_7498_transp__empty,axiom,
! [A: $tType] :
( transp @ A
@ ^ [X4: A,Y3: A] : $false ) ).
% transp_empty
thf(fact_7499_transp__singleton,axiom,
! [A: $tType,A3: A] :
( transp @ A
@ ^ [X4: A,Y3: A] :
( ( X4 = A3 )
& ( Y3 = A3 ) ) ) ).
% transp_singleton
thf(fact_7500_transp__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less @ A ) ) ) ).
% transp_less
thf(fact_7501_transp__gr,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X4: A,Y3: A] : ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% transp_gr
thf(fact_7502_transp__ge,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X4: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% transp_ge
thf(fact_7503_transp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less_eq @ A ) ) ) ).
% transp_le
thf(fact_7504_transp__INF,axiom,
! [B: $tType,A: $tType,S3: set @ A,R2: A > B > B > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( transp @ B @ ( R2 @ X5 ) ) )
=> ( transp @ B @ ( complete_Inf_Inf @ ( B > B > $o ) @ ( image2 @ A @ ( B > B > $o ) @ R2 @ S3 ) ) ) ) ).
% transp_INF
thf(fact_7505_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ B ),F4: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y3: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y3 ) ) @ R5 ) ) ) ) ) ).
% inv_image_def
thf(fact_7506_rp__inv__image__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_rp_inv_image @ A @ B )
= ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
@ ^ [R6: set @ ( product_prod @ A @ A ),S6: set @ ( product_prod @ A @ A ),F4: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R6 @ F4 ) @ ( inv_image @ A @ B @ S6 @ F4 ) ) ) ) ).
% rp_inv_image_def
thf(fact_7507_lenlex__def,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ ( list @ A ) ) @ ( list @ A ) @ ( lex_prod @ nat @ ( list @ A ) @ less_than @ ( lex @ A @ R5 ) )
@ ^ [Xs: list @ A] : ( product_Pair @ nat @ ( list @ A ) @ ( size_size @ ( list @ A ) @ Xs ) @ Xs ) ) ) ) ).
% lenlex_def
thf(fact_7508_less__than__iff,axiom,
! [X3: nat,Y: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X3 @ Y ) @ less_than )
= ( ord_less @ nat @ X3 @ Y ) ) ).
% less_than_iff
thf(fact_7509_mlex__prod__def,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F4: A > nat,R6: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ A ) @ A @ ( lex_prod @ nat @ A @ less_than @ R6 )
@ ^ [X4: A] : ( product_Pair @ nat @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ).
% mlex_prod_def
thf(fact_7510_bsqr__max2,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A1: A,A22: A,B1: A,B22: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) @ ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( product_Pair @ A @ A @ B1 @ B22 ) ) @ ( bNF_Wellorder_bsqr @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A1 @ A22 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ B1 @ B22 ) ) @ R2 ) ) ) ).
% bsqr_max2
thf(fact_7511_lexordp__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% lexordp_conv_lexord
thf(fact_7512_well__order__on__domain,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_well_order_on @ A @ A6 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( ( member @ A @ A3 @ A6 )
& ( member @ A @ B2 @ A6 ) ) ) ) ).
% well_order_on_domain
thf(fact_7513_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_7514_natLeq__on__well__order__on,axiom,
! [N: nat] :
( order_well_order_on @ nat
@ ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y3: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X4 @ Y3 ) ) ) ) ) ).
% natLeq_on_well_order_on
thf(fact_7515_well__order__on__Restr,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( field2 @ A @ R2 ) )
=> ( order_well_order_on @ A @ A6
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu3: A] : A6 ) ) ) ) ) ).
% well_order_on_Restr
thf(fact_7516_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
@ ^ [X4: nat,Y3: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X4 @ Y3 ) ) ) ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y3: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X4 @ Y3 ) ) ) ) ) ).
% natLeq_on_Well_order
thf(fact_7517_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 )
= ( ! [A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R2 ) )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_7518_ofilter__Restr__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A,B5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( order_ofilter @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B5
@ ^ [Uu3: A] : B5 ) )
@ A6 ) ) ) ) ).
% ofilter_Restr_subset
thf(fact_7519_ofilter__subset__ordLess,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A,B5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A6 )
=> ( ( order_ofilter @ A @ R2 @ B5 )
=> ( ( ord_less @ ( set @ A ) @ A6 @ B5 )
= ( 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 @ A6
@ ^ [Uu3: A] : A6 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B5
@ ^ [Uu3: A] : B5 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLess
thf(fact_7520_ofilter__ordLess,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A6 )
=> ( ( ord_less @ ( set @ A ) @ A6 @ ( 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 @ A6
@ ^ [Uu3: A] : A6 ) )
@ R2 )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ).
% ofilter_ordLess
thf(fact_7521_ordLess__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R7: 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 ) ) @ R2 @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_transitive
thf(fact_7522_ordLess__irreflexive,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 @ R2 ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ).
% ordLess_irreflexive
thf(fact_7523_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_7524_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_7525_ordLess__iff__ordIso__Restr,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 )
=> ( ( 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 ) ) @ R4 @ R2 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ ( 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 ) ) @ R4
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R2 @ X4 )
@ ^ [Uu3: A] : ( order_underS @ A @ R2 @ X4 ) ) ) )
@ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ) ) ) ) ).
% ordLess_iff_ordIso_Restr
thf(fact_7526_ordLeq__iff__ordLess__Restr,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 )
=> ( ( 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_Wellorder_ordLeq @ A @ B ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ 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 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R2 @ X4 )
@ ^ [Uu3: A] : ( order_underS @ A @ R2 @ X4 ) ) )
@ R4 )
@ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ) ).
% ordLeq_iff_ordLess_Restr
thf(fact_7527_ordLeq__Well__order__simp,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R4: 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
& ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 ) ) ) ).
% ordLeq_Well_order_simp
thf(fact_7528_ordLeq__reflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ 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 ) ) @ R2 @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% ordLeq_reflexive
thf(fact_7529_ordLeq__total,axiom,
! [A: $tType,B: $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 )
=> ( ( 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_Wellorder_ordLeq @ A @ 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 ) ) @ R4 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% ordLeq_total
thf(fact_7530_exists__minim__Well__order,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X5: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X5 @ R )
=> ( order_well_order_on @ A @ ( field2 @ A @ X5 ) @ X5 ) )
=> ? [X5: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X5 @ R )
& ! [Xa: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa @ R )
=> ( 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 ) ) @ X5 @ Xa ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Well_order
thf(fact_7531_finite__well__order__on__ordIso,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A6 )
=> ( ( order_well_order_on @ A @ A6 @ R2 )
=> ( ( order_well_order_on @ A @ A6 @ 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_7532_not__ordLess__ordIso,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: 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 ) ) @ R2 @ R4 ) @ ( bNF_We4044943003108391690rdLess @ 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% not_ordLess_ordIso
thf(fact_7533_not__ordLess__ordLeq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R4: 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 ) ) @ R2 @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ 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 ) ) @ R4 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% not_ordLess_ordLeq
thf(fact_7534_ordLess__imp__ordLeq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: 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 ) ) @ R2 @ R4 ) @ ( bNF_We4044943003108391690rdLess @ 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% ordLess_imp_ordLeq
thf(fact_7535_ordIso__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R7: 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordIso_ordLess_trans
thf(fact_7536_ordLeq__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R7: 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLeq_ordLess_trans
thf(fact_7537_ordLess__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R7: 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 ) ) @ R2 @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R7 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_ordIso_trans
thf(fact_7538_ordLess__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R7: 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 ) ) @ R2 @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R7 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_ordLeq_trans
thf(fact_7539_ordLeq__iff__ordLess__or__ordIso,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: 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 ) ) @ R2 @ R4 ) @ ( 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 ) ) @ R2 @ R4 ) @ ( bNF_We4044943003108391690rdLess @ 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% ordLeq_iff_ordLess_or_ordIso
thf(fact_7540_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_7541_ordLeq__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R7: 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R7 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordLeq_ordIso_trans
thf(fact_7542_ordIso__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R7: 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R7 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordIso_ordLeq_trans
thf(fact_7543_ordLeq__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R7: 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R7 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordLeq_transitive
thf(fact_7544_ordIso__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R7: 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R7 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ C ) ) ) ) ).
% ordIso_transitive
thf(fact_7545_ordIso__imp__ordLeq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordIso @ 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% ordIso_imp_ordLeq
thf(fact_7546_ordIso__iff__ordLeq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R4: 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordIso @ 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ 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 ) ) @ R4 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% ordIso_iff_ordLeq
thf(fact_7547_ordIso__symmetric,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R4: 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ 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 ) ) @ R4 @ R2 ) @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ).
% ordIso_symmetric
thf(fact_7548_ordIso__reflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ 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 ) ) @ R2 @ R2 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% ordIso_reflexive
thf(fact_7549_Well__order__iso__copy,axiom,
! [B: $tType,A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > B,A11: set @ B] :
( ( order_well_order_on @ A @ A6 @ R2 )
=> ( ( bij_betw @ A @ B @ F3 @ A6 @ A11 )
=> ? [R8: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ B @ A11 @ R8 )
& ( 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 @ R8 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% Well_order_iso_copy
thf(fact_7550_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_7551_ordLess__or__ordLeq,axiom,
! [A: $tType,B: $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 )
=> ( ( 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 ) )
| ( 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 ) ) @ R4 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% ordLess_or_ordLeq
thf(fact_7552_not__ordLeq__iff__ordLess,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 )
=> ( ( ~ ( 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 ) ) @ R4 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ 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 @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ).
% not_ordLeq_iff_ordLess
thf(fact_7553_not__ordLess__iff__ordLeq,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 )
=> ( ( ~ ( 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 ) ) @ R4 @ R2 ) @ ( bNF_We4044943003108391690rdLess @ B @ 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 @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% not_ordLess_iff_ordLeq
thf(fact_7554_ofilter__subset__ordLeq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A,B5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A6 )
=> ( ( order_ofilter @ A @ R2 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
= ( 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 @ A6
@ ^ [Uu3: A] : A6 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B5
@ ^ [Uu3: A] : B5 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLeq
thf(fact_7555_dir__image__ordIso,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F3: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( inj_on @ A @ B @ F3 @ ( field2 @ A @ 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_We2720479622203943262_image @ A @ B @ R2 @ F3 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% dir_image_ordIso
thf(fact_7556_bounded__bilinear__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ( ( real_V2442710119149674383linear @ A @ B @ C )
= ( ^ [Prod3: A > B > C] :
( ! [A8: A,A14: A,B8: B] :
( ( Prod3 @ ( plus_plus @ A @ A8 @ A14 ) @ B8 )
= ( plus_plus @ C @ ( Prod3 @ A8 @ B8 ) @ ( Prod3 @ A14 @ B8 ) ) )
& ! [A8: A,B8: B,B12: B] :
( ( Prod3 @ A8 @ ( plus_plus @ B @ B8 @ B12 ) )
= ( plus_plus @ C @ ( Prod3 @ A8 @ B8 ) @ ( Prod3 @ A8 @ B12 ) ) )
& ! [R5: real,A8: A,B8: B] :
( ( Prod3 @ ( real_V8093663219630862766scaleR @ A @ R5 @ A8 ) @ B8 )
= ( real_V8093663219630862766scaleR @ C @ R5 @ ( Prod3 @ A8 @ B8 ) ) )
& ! [A8: A,R5: real,B8: B] :
( ( Prod3 @ A8 @ ( real_V8093663219630862766scaleR @ B @ R5 @ B8 ) )
= ( real_V8093663219630862766scaleR @ C @ R5 @ ( Prod3 @ A8 @ B8 ) ) )
& ? [K6: real] :
! [A8: A,B8: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod3 @ A8 @ B8 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A8 ) @ ( real_V7770717601297561774m_norm @ B @ B8 ) ) @ K6 ) ) ) ) ) ) ).
% bounded_bilinear_def
thf(fact_7557_bounded__bilinear_Oprod__diff__prod,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A ) )
=> ! [Prod: A > B > C,X3: A,Y: B,A3: A,B2: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( minus_minus @ C @ ( Prod @ X3 @ Y ) @ ( Prod @ A3 @ B2 ) )
= ( plus_plus @ C @ ( plus_plus @ C @ ( Prod @ ( minus_minus @ A @ X3 @ A3 ) @ ( minus_minus @ B @ Y @ B2 ) ) @ ( Prod @ ( minus_minus @ A @ X3 @ A3 ) @ B2 ) ) @ ( Prod @ A3 @ ( minus_minus @ B @ Y @ B2 ) ) ) ) ) ) ).
% bounded_bilinear.prod_diff_prod
thf(fact_7558_bounded__bilinear_Oadd__left,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A ) )
=> ! [Prod: A > B > C,A3: A,A4: A,B2: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( Prod @ ( plus_plus @ A @ A3 @ A4 ) @ B2 )
= ( plus_plus @ C @ ( Prod @ A3 @ B2 ) @ ( Prod @ A4 @ B2 ) ) ) ) ) ).
% bounded_bilinear.add_left
thf(fact_7559_bounded__bilinear_Oadd__right,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A ) )
=> ! [Prod: A > B > C,A3: A,B2: B,B3: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( Prod @ A3 @ ( plus_plus @ B @ B2 @ B3 ) )
= ( plus_plus @ C @ ( Prod @ A3 @ B2 ) @ ( Prod @ A3 @ B3 ) ) ) ) ) ).
% bounded_bilinear.add_right
thf(fact_7560_dir__image__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_We2720479622203943262_image @ A @ A2 )
= ( ^ [R5: set @ ( product_prod @ A @ A ),F4: A > A2] :
( collect @ ( product_prod @ A2 @ A2 )
@ ^ [Uu3: product_prod @ A2 @ A2] :
? [A8: A,B8: A] :
( ( Uu3
= ( product_Pair @ A2 @ A2 @ ( F4 @ A8 ) @ ( F4 @ B8 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B8 ) @ R5 ) ) ) ) ) ).
% dir_image_def
thf(fact_7561_bounded__bilinear_OFDERIV,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ D )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A ) )
=> ! [Prod: A > B > C,F3: D > A,F8: D > A,X3: D,S: set @ D,G3: D > B,G6: D > B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( has_derivative @ D @ A @ F3 @ F8 @ ( topolo174197925503356063within @ D @ X3 @ S ) )
=> ( ( has_derivative @ D @ B @ G3 @ G6 @ ( topolo174197925503356063within @ D @ X3 @ S ) )
=> ( has_derivative @ D @ C
@ ^ [X4: D] : ( Prod @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ^ [H2: D] : ( plus_plus @ C @ ( Prod @ ( F3 @ X3 ) @ ( G6 @ H2 ) ) @ ( Prod @ ( F8 @ H2 ) @ ( G3 @ X3 ) ) )
@ ( topolo174197925503356063within @ D @ X3 @ S ) ) ) ) ) ) ).
% bounded_bilinear.FDERIV
thf(fact_7562_bounded__bilinear_Ointro,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C] :
( ! [A5: A,A21: A,B4: B] :
( ( Prod @ ( plus_plus @ A @ A5 @ A21 ) @ B4 )
= ( plus_plus @ C @ ( Prod @ A5 @ B4 ) @ ( Prod @ A21 @ B4 ) ) )
=> ( ! [A5: A,B4: B,B16: B] :
( ( Prod @ A5 @ ( plus_plus @ B @ B4 @ B16 ) )
= ( plus_plus @ C @ ( Prod @ A5 @ B4 ) @ ( Prod @ A5 @ B16 ) ) )
=> ( ! [R3: real,A5: A,B4: B] :
( ( Prod @ ( real_V8093663219630862766scaleR @ A @ R3 @ A5 ) @ B4 )
= ( real_V8093663219630862766scaleR @ C @ R3 @ ( Prod @ A5 @ B4 ) ) )
=> ( ! [A5: A,R3: real,B4: B] :
( ( Prod @ A5 @ ( real_V8093663219630862766scaleR @ B @ R3 @ B4 ) )
= ( real_V8093663219630862766scaleR @ C @ R3 @ ( Prod @ A5 @ B4 ) ) )
=> ( ? [K8: real] :
! [A5: A,B4: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod @ A5 @ B4 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A5 ) @ ( real_V7770717601297561774m_norm @ B @ B4 ) ) @ K8 ) )
=> ( real_V2442710119149674383linear @ A @ B @ C @ Prod ) ) ) ) ) ) ) ).
% bounded_bilinear.intro
thf(fact_7563_coinduct3__lemma,axiom,
! [A: $tType,X6: set @ A,F3: ( set @ A ) > ( set @ A )] :
( ( ord_less_eq @ ( set @ A ) @ X6
@ ( F3
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X4: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F3 @ X4 ) @ X6 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ord_less_eq @ ( set @ A )
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X4: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F3 @ X4 ) @ X6 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) )
@ ( F3
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X4: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F3 @ X4 ) @ X6 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) ) ) ) ).
% coinduct3_lemma
thf(fact_7564_def__coinduct3,axiom,
! [A: $tType,A6: set @ A,F3: ( set @ A ) > ( set @ A ),A3: A,X6: set @ A] :
( ( A6
= ( complete_lattice_gfp @ ( set @ A ) @ F3 ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ( member @ A @ A3 @ X6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6
@ ( F3
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X4: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F3 @ X4 ) @ X6 ) @ A6 ) ) ) )
=> ( member @ A @ A3 @ A6 ) ) ) ) ) ).
% def_coinduct3
thf(fact_7565_gfp__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F6: A > A,X3: A] :
( ( order_mono @ A @ A @ F6 )
=> ( ( ( F6 @ X3 )
= X3 )
=> ( ! [Z3: A] :
( ( ( F6 @ Z3 )
= Z3 )
=> ( ord_less_eq @ A @ Z3 @ X3 ) )
=> ( ( complete_lattice_gfp @ A @ F6 )
= X3 ) ) ) ) ) ).
% gfp_eqI
thf(fact_7566_weak__coinduct,axiom,
! [A: $tType,A3: A,X6: set @ A,F3: ( set @ A ) > ( set @ A )] :
( ( member @ A @ A3 @ X6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ ( F3 @ X6 ) )
=> ( member @ A @ A3 @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) ).
% weak_coinduct
thf(fact_7567_gfp__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,G3: A > A] :
( ! [Z10: A] : ( ord_less_eq @ A @ ( F3 @ Z10 ) @ ( G3 @ Z10 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F3 ) @ ( complete_lattice_gfp @ A @ G3 ) ) ) ) ).
% gfp_mono
thf(fact_7568_gfp__upperbound,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X6: A,F3: A > A] :
( ( ord_less_eq @ A @ X6 @ ( F3 @ X6 ) )
=> ( ord_less_eq @ A @ X6 @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ).
% gfp_upperbound
thf(fact_7569_gfp__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,X6: A] :
( ! [U4: A] :
( ( ord_less_eq @ A @ U4 @ ( F3 @ U4 ) )
=> ( ord_less_eq @ A @ U4 @ X6 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F3 ) @ X6 ) ) ) ).
% gfp_least
thf(fact_7570_gfp__gfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A > A] :
( ! [X5: A,Y4: A,W2: A,Z3: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ( ord_less_eq @ A @ W2 @ Z3 )
=> ( ord_less_eq @ A @ ( F3 @ X5 @ W2 ) @ ( F3 @ Y4 @ Z3 ) ) ) )
=> ( ( complete_lattice_gfp @ A
@ ^ [X4: A] : ( complete_lattice_gfp @ A @ ( F3 @ X4 ) ) )
= ( complete_lattice_gfp @ A
@ ^ [X4: A] : ( F3 @ X4 @ X4 ) ) ) ) ) ).
% gfp_gfp
thf(fact_7571_gfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_gfp @ A )
= ( ^ [F4: A > A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [U2: A] : ( ord_less_eq @ A @ U2 @ ( F4 @ U2 ) ) ) ) ) ) ) ).
% gfp_def
thf(fact_7572_weak__coinduct__image,axiom,
! [A: $tType,B: $tType,A3: A,X6: set @ A,G3: A > B,F3: ( set @ B ) > ( set @ B )] :
( ( member @ A @ A3 @ X6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G3 @ X6 ) @ ( F3 @ ( image2 @ A @ B @ G3 @ X6 ) ) )
=> ( member @ B @ ( G3 @ A3 ) @ ( complete_lattice_gfp @ ( set @ B ) @ F3 ) ) ) ) ).
% weak_coinduct_image
thf(fact_7573_coinduct__lemma,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X6: A,F3: A > A] :
( ( ord_less_eq @ A @ X6 @ ( F3 @ ( sup_sup @ A @ X6 @ ( complete_lattice_gfp @ A @ F3 ) ) ) )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ X6 @ ( complete_lattice_gfp @ A @ F3 ) ) @ ( F3 @ ( sup_sup @ A @ X6 @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ) ) ) ).
% coinduct_lemma
thf(fact_7574_def__coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: A,F3: A > A,X6: A] :
( ( A6
= ( complete_lattice_gfp @ A @ F3 ) )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ X6 @ ( F3 @ ( sup_sup @ A @ X6 @ A6 ) ) )
=> ( ord_less_eq @ A @ X6 @ A6 ) ) ) ) ) ).
% def_coinduct
thf(fact_7575_coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,X6: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ X6 @ ( F3 @ ( sup_sup @ A @ X6 @ ( complete_lattice_gfp @ A @ F3 ) ) ) )
=> ( ord_less_eq @ A @ X6 @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ) ).
% coinduct
thf(fact_7576_def__coinduct__set,axiom,
! [A: $tType,A6: set @ A,F3: ( set @ A ) > ( set @ A ),A3: A,X6: set @ A] :
( ( A6
= ( complete_lattice_gfp @ ( set @ A ) @ F3 ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ( member @ A @ A3 @ X6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ ( F3 @ ( sup_sup @ ( set @ A ) @ X6 @ A6 ) ) )
=> ( member @ A @ A3 @ A6 ) ) ) ) ) ).
% def_coinduct_set
thf(fact_7577_coinduct__set,axiom,
! [A: $tType,F3: ( set @ A ) > ( set @ A ),A3: A,X6: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ( member @ A @ A3 @ X6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ ( F3 @ ( sup_sup @ ( set @ A ) @ X6 @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) )
=> ( member @ A @ A3 @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) ) ).
% coinduct_set
thf(fact_7578_gfp__ordinal__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,P: A > $o] :
( ( order_mono @ A @ A @ F3 )
=> ( ! [S4: A] :
( ( P @ S4 )
=> ( ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F3 ) @ S4 )
=> ( P @ ( F3 @ S4 ) ) ) )
=> ( ! [M8: set @ A] :
( ! [X: A] :
( ( member @ A @ X @ M8 )
=> ( P @ X ) )
=> ( P @ ( complete_Inf_Inf @ A @ M8 ) ) )
=> ( P @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ) ) ).
% gfp_ordinal_induct
thf(fact_7579_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_7580_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_7581_gfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,K2: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K2 ) @ F3 @ ( top_top @ A ) )
= ( compow @ ( A > A ) @ K2 @ F3 @ ( top_top @ A ) ) )
=> ( ( complete_lattice_gfp @ A @ F3 )
= ( compow @ ( A > A ) @ K2 @ F3 @ ( top_top @ A ) ) ) ) ) ) ).
% gfp_Kleene_iter
thf(fact_7582_coinduct3,axiom,
! [A: $tType,F3: ( set @ A ) > ( set @ A ),A3: A,X6: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ( member @ A @ A3 @ X6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6
@ ( F3
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X4: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F3 @ X4 ) @ X6 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) )
=> ( member @ A @ A3 @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) ) ).
% coinduct3
thf(fact_7583_gfp__transfer__bounded,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [P: A > $o,F3: A > A,Alpha: A > B,G3: B > B] :
( ( P @ ( F3 @ ( top_top @ A ) ) )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( P @ ( F3 @ X5 ) ) )
=> ( ! [M8: nat > A] :
( ( order_antimono @ nat @ A @ M8 )
=> ( ! [I2: nat] : ( P @ ( M8 @ I2 ) )
=> ( P @ ( complete_Inf_Inf @ A @ ( image2 @ nat @ A @ M8 @ ( top_top @ ( set @ nat ) ) ) ) ) ) )
=> ( ! [M8: nat > A] :
( ( order_antimono @ nat @ A @ M8 )
=> ( ! [I2: nat] : ( P @ ( M8 @ I2 ) )
=> ( ( Alpha @ ( complete_Inf_Inf @ A @ ( image2 @ nat @ A @ M8 @ ( top_top @ ( set @ nat ) ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image2 @ nat @ B
@ ^ [I4: nat] : ( Alpha @ ( M8 @ I4 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) )
=> ( ( order_inf_continuous @ A @ A @ F3 )
=> ( ( order_inf_continuous @ B @ B @ G3 )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( ( Alpha @ ( F3 @ X5 ) )
= ( G3 @ ( Alpha @ X5 ) ) ) )
=> ( ! [X5: B] : ( ord_less_eq @ B @ ( G3 @ X5 ) @ ( Alpha @ ( F3 @ ( top_top @ A ) ) ) )
=> ( ( Alpha @ ( complete_lattice_gfp @ A @ F3 ) )
= ( complete_lattice_gfp @ B @ G3 ) ) ) ) ) ) ) ) ) ) ) ).
% gfp_transfer_bounded
thf(fact_7584_prod__decode__triangle__add,axiom,
! [K2: nat,M2: nat] :
( ( nat_prod_decode @ ( plus_plus @ nat @ ( nat_triangle @ K2 ) @ M2 ) )
= ( nat_prod_decode_aux @ K2 @ M2 ) ) ).
% prod_decode_triangle_add
thf(fact_7585_prod__decode__eq,axiom,
! [X3: nat,Y: nat] :
( ( ( nat_prod_decode @ X3 )
= ( nat_prod_decode @ Y ) )
= ( X3 = Y ) ) ).
% prod_decode_eq
thf(fact_7586_prod__decode__inverse,axiom,
! [N: nat] :
( ( nat_prod_encode @ ( nat_prod_decode @ N ) )
= N ) ).
% prod_decode_inverse
thf(fact_7587_prod__encode__inverse,axiom,
! [X3: product_prod @ nat @ nat] :
( ( nat_prod_decode @ ( nat_prod_encode @ X3 ) )
= X3 ) ).
% prod_encode_inverse
thf(fact_7588_inj__prod__decode,axiom,
! [A6: set @ nat] : ( inj_on @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ A6 ) ).
% inj_prod_decode
thf(fact_7589_prod__decode__def,axiom,
( nat_prod_decode
= ( nat_prod_decode_aux @ ( zero_zero @ nat ) ) ) ).
% prod_decode_def
thf(fact_7590_bij__prod__decode,axiom,
bij_betw @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% bij_prod_decode
thf(fact_7591_surj__prod__decode,axiom,
( ( image2 @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ).
% surj_prod_decode
thf(fact_7592_list__decode_Opinduct,axiom,
! [A0: nat,P: nat > $o] :
( ( accp @ nat @ nat_list_decode_rel @ A0 )
=> ( ( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
=> ( ! [N2: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N2 ) )
=> ( ! [X: nat,Y6: nat] :
( ( ( product_Pair @ nat @ nat @ X @ Y6 )
= ( nat_prod_decode @ N2 ) )
=> ( P @ Y6 ) )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% list_decode.pinduct
thf(fact_7593_list__decode_Oelims,axiom,
! [X3: nat,Y: list @ nat] :
( ( ( nat_list_decode @ X3 )
= Y )
=> ( ( ( X3
= ( zero_zero @ nat ) )
=> ( Y
!= ( nil @ nat ) ) )
=> ~ ! [N2: nat] :
( ( X3
= ( suc @ N2 ) )
=> ( Y
!= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X4: nat,Y3: nat] : ( cons @ nat @ X4 @ ( nat_list_decode @ Y3 ) )
@ ( nat_prod_decode @ N2 ) ) ) ) ) ) ).
% list_decode.elims
thf(fact_7594_list__encode__inverse,axiom,
! [X3: list @ nat] :
( ( nat_list_decode @ ( nat_list_encode @ X3 ) )
= X3 ) ).
% list_encode_inverse
thf(fact_7595_list__decode__inverse,axiom,
! [N: nat] :
( ( nat_list_encode @ ( nat_list_decode @ N ) )
= N ) ).
% list_decode_inverse
thf(fact_7596_list__decode_Opsimps_I1_J,axiom,
( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
=> ( ( nat_list_decode @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ) ).
% list_decode.psimps(1)
thf(fact_7597_inj__list__decode,axiom,
! [A6: set @ nat] : ( inj_on @ nat @ ( list @ nat ) @ nat_list_decode @ A6 ) ).
% inj_list_decode
thf(fact_7598_list__decode__eq,axiom,
! [X3: nat,Y: nat] :
( ( ( nat_list_decode @ X3 )
= ( nat_list_decode @ Y ) )
= ( X3 = Y ) ) ).
% list_decode_eq
thf(fact_7599_list__decode_Osimps_I1_J,axiom,
( ( nat_list_decode @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% list_decode.simps(1)
thf(fact_7600_list__decode_Opsimps_I2_J,axiom,
! [N: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N ) )
=> ( ( nat_list_decode @ ( suc @ N ) )
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X4: nat,Y3: nat] : ( cons @ nat @ X4 @ ( nat_list_decode @ Y3 ) )
@ ( nat_prod_decode @ N ) ) ) ) ).
% list_decode.psimps(2)
thf(fact_7601_bij__list__decode,axiom,
bij_betw @ nat @ ( list @ nat ) @ nat_list_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( list @ nat ) ) ) ).
% bij_list_decode
thf(fact_7602_surj__list__decode,axiom,
( ( image2 @ nat @ ( list @ nat ) @ nat_list_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ ( list @ nat ) ) ) ) ).
% surj_list_decode
thf(fact_7603_list__decode_Osimps_I2_J,axiom,
! [N: nat] :
( ( nat_list_decode @ ( suc @ N ) )
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X4: nat,Y3: nat] : ( cons @ nat @ X4 @ ( nat_list_decode @ Y3 ) )
@ ( nat_prod_decode @ N ) ) ) ).
% list_decode.simps(2)
thf(fact_7604_list__decode_Opelims,axiom,
! [X3: nat,Y: list @ nat] :
( ( ( nat_list_decode @ X3 )
= Y )
=> ( ( accp @ nat @ nat_list_decode_rel @ X3 )
=> ( ( ( X3
= ( zero_zero @ nat ) )
=> ( ( Y
= ( nil @ nat ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N2: nat] :
( ( X3
= ( suc @ N2 ) )
=> ( ( Y
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X4: nat,Y3: nat] : ( cons @ nat @ X4 @ ( nat_list_decode @ Y3 ) )
@ ( nat_prod_decode @ N2 ) ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( suc @ N2 ) ) ) ) ) ) ) ).
% list_decode.pelims
thf(fact_7605_surj__swap,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% surj_swap
thf(fact_7606_Max_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_max @ A )
@ ^ [X4: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X4 )
@ ^ [X4: A,Y3: A] : ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% Max.semilattice_order_set_axioms
thf(fact_7607_swap__swap,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
= P2 ) ).
% swap_swap
thf(fact_7608_swap__simp,axiom,
! [A: $tType,B: $tType,X3: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) )
= ( product_Pair @ A @ B @ Y @ X3 ) ) ).
% swap_simp
thf(fact_7609_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > B > A,P2: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y3: B,X4: C] : ( F3 @ X4 @ Y3 )
@ ( product_swap @ C @ B @ P2 ) )
= ( product_case_prod @ C @ B @ A @ F3 @ P2 ) ) ).
% case_swap
thf(fact_7610_fst__swap,axiom,
! [A: $tType,B: $tType,X3: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X3 ) )
= ( product_snd @ B @ A @ X3 ) ) ).
% fst_swap
thf(fact_7611_snd__swap,axiom,
! [B: $tType,A: $tType,X3: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X3 ) )
= ( product_fst @ A @ B @ X3 ) ) ).
% snd_swap
thf(fact_7612_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X3: B,A6: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X3 ) @ ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A6 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y ) @ A6 ) ) ).
% pair_in_swap_image
thf(fact_7613_inj__swap,axiom,
! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A6 ) ).
% inj_swap
thf(fact_7614_bij__swap,axiom,
! [A: $tType,B: $tType] : ( bij_betw @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) ) ).
% bij_swap
thf(fact_7615_product__swap,axiom,
! [B: $tType,A: $tType,A6: set @ B,B5: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu3: B] : B5 ) )
= ( product_Sigma @ A @ B @ B5
@ ^ [Uu3: A] : A6 ) ) ).
% product_swap
thf(fact_7616_prod__filter__commute,axiom,
! [B: $tType,A: $tType] :
( ( prod_filter @ A @ B )
= ( ^ [F9: filter @ A,G9: filter @ B] : ( filtermap @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( prod_filter @ B @ A @ G9 @ F9 ) ) ) ) ).
% prod_filter_commute
thf(fact_7617_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_7618_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P5: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P5 ) @ ( product_fst @ A @ B @ P5 ) ) ) ) ).
% prod.swap_def
thf(fact_7619_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_7620_Sup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic4895041142388067077er_set @ A @ ( sup_sup @ A )
@ ^ [X4: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X4 )
@ ^ [X4: A,Y3: A] : ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_7621_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X3: A > B,Xa2: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X3 @ Xa2 ) )
=> ( ! [X5: A] :
( ( Xa2
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ( Y = X5 )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X3 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) ) )
=> ( ! [X5: A,Y4: A,Zs: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Zs ) ) )
=> ( ( Y
= ( if @ A @ ( ord_less_eq @ B @ ( X3 @ X5 ) @ ( X3 @ ( arg_min_list @ A @ B @ X3 @ ( cons @ A @ Y4 @ Zs ) ) ) ) @ X5 @ ( arg_min_list @ A @ B @ X3 @ ( cons @ A @ Y4 @ Zs ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X3 @ ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Zs ) ) ) ) ) )
=> ~ ( ( Xa2
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X3 @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% arg_min_list.pelims
thf(fact_7622_flip__pred,axiom,
! [A: $tType,B: $tType,A6: set @ ( product_prod @ A @ B ),R: B > A > $o] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( conversep @ B @ A @ R ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X4: A,Y3: B] : ( product_Pair @ B @ A @ Y3 @ X4 ) )
@ A6 )
@ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ R ) ) ) ) ).
% flip_pred
thf(fact_7623_conversep__eq,axiom,
! [A: $tType] :
( ( conversep @ A @ A
@ ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [Y5: A,Z: A] : Y5 = Z ) ) ).
% conversep_eq
thf(fact_7624_conversep__iff,axiom,
! [B: $tType,A: $tType] :
( ( conversep @ A @ B )
= ( ^ [R5: A > B > $o,A8: B,B8: A] : ( R5 @ B8 @ A8 ) ) ) ).
% conversep_iff
thf(fact_7625_conversep__inject,axiom,
! [A: $tType,B: $tType,R2: B > A > $o,S: B > A > $o] :
( ( ( conversep @ B @ A @ R2 )
= ( conversep @ B @ A @ S ) )
= ( R2 = S ) ) ).
% conversep_inject
thf(fact_7626_conversep__conversep,axiom,
! [B: $tType,A: $tType,R2: A > B > $o] :
( ( conversep @ B @ A @ ( conversep @ A @ B @ R2 ) )
= R2 ) ).
% conversep_conversep
thf(fact_7627_conversep__noteq,axiom,
! [A: $tType] :
( ( conversep @ A @ A
@ ^ [X4: A,Y3: A] : X4 != Y3 )
= ( ^ [X4: A,Y3: A] : X4 != Y3 ) ) ).
% conversep_noteq
thf(fact_7628_conversep__mono,axiom,
! [A: $tType,B: $tType,R2: B > A > $o,S: B > A > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ ( conversep @ B @ A @ R2 ) @ ( conversep @ B @ A @ S ) )
= ( ord_less_eq @ ( B > A > $o ) @ R2 @ S ) ) ).
% conversep_mono
thf(fact_7629_converse__meet,axiom,
! [A: $tType,B: $tType,R2: B > A > $o,S: B > A > $o] :
( ( conversep @ B @ A @ ( inf_inf @ ( B > A > $o ) @ R2 @ S ) )
= ( inf_inf @ ( A > B > $o ) @ ( conversep @ B @ A @ R2 ) @ ( conversep @ B @ A @ S ) ) ) ).
% converse_meet
thf(fact_7630_conversep_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( conversep @ A @ B )
= ( ^ [R5: A > B > $o,A12: B,A23: A] :
? [A8: A,B8: B] :
( ( A12 = B8 )
& ( A23 = A8 )
& ( R5 @ A8 @ B8 ) ) ) ) ).
% conversep.simps
thf(fact_7631_conversepD,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,B2: B,A3: A] :
( ( conversep @ A @ B @ R2 @ B2 @ A3 )
=> ( R2 @ A3 @ B2 ) ) ).
% conversepD
thf(fact_7632_conversepE,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A1: B,A22: A] :
( ( conversep @ A @ B @ R2 @ A1 @ A22 )
=> ( R2 @ A22 @ A1 ) ) ).
% conversepE
thf(fact_7633_conversepI,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,A3: A,B2: B] :
( ( R2 @ A3 @ B2 )
=> ( conversep @ A @ B @ R2 @ B2 @ A3 ) ) ).
% conversepI
thf(fact_7634_converse__join,axiom,
! [A: $tType,B: $tType,R2: B > A > $o,S: B > A > $o] :
( ( conversep @ B @ A @ ( sup_sup @ ( B > A > $o ) @ R2 @ S ) )
= ( sup_sup @ ( A > B > $o ) @ ( conversep @ B @ A @ R2 ) @ ( conversep @ B @ A @ S ) ) ) ).
% converse_join
thf(fact_7635_conversep__le__swap,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,S: B > A > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R2 @ ( conversep @ B @ A @ S ) )
= ( ord_less_eq @ ( B > A > $o ) @ ( conversep @ A @ B @ R2 ) @ S ) ) ).
% conversep_le_swap
thf(fact_7636_sorted__wrt_Opelims_I2_J,axiom,
! [A: $tType,X3: A > A > $o,Xa2: list @ A] :
( ( sorted_wrt @ A @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ Xa2 ) )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( nil @ A ) ) ) )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ Ys4 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( cons @ A @ X5 @ Ys4 ) ) )
=> ~ ( ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) )
=> ( X3 @ X5 @ Xa ) )
& ( sorted_wrt @ A @ X3 @ Ys4 ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(2)
thf(fact_7637_sorted__wrt_Opelims_I1_J,axiom,
! [A: $tType,X3: A > A > $o,Xa2: list @ A,Y: $o] :
( ( ( sorted_wrt @ A @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ Xa2 ) )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( nil @ A ) ) ) ) )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ Ys4 ) )
=> ( ( Y
= ( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Ys4 ) )
=> ( X3 @ X5 @ Y3 ) )
& ( sorted_wrt @ A @ X3 @ Ys4 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( cons @ A @ X5 @ Ys4 ) ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(1)
thf(fact_7638_sorted__wrt_Opelims_I3_J,axiom,
! [A: $tType,X3: A > A > $o,Xa2: list @ A] :
( ~ ( sorted_wrt @ A @ X3 @ Xa2 )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ Xa2 ) )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ Ys4 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( cons @ A @ X5 @ Ys4 ) ) )
=> ( ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Ys4 ) )
=> ( X3 @ X5 @ Xa3 ) )
& ( sorted_wrt @ A @ X3 @ Ys4 ) ) ) ) ) ) ).
% sorted_wrt.pelims(3)
thf(fact_7639_ex__is__arg__min__if__finite,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S3: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_12: A] :
( lattic501386751177426532rg_min @ A @ B @ F3
@ ^ [X4: A] : ( member @ A @ X4 @ S3 )
@ X_12 ) ) ) ) ).
% ex_is_arg_min_if_finite
thf(fact_7640_map__comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( map_comp @ B @ C @ A )
= ( ^ [F4: B > ( option @ C ),G4: A > ( option @ B ),K3: A] : ( case_option @ ( option @ C ) @ B @ ( none @ C ) @ F4 @ ( G4 @ K3 ) ) ) ) ).
% map_comp_def
thf(fact_7641_map__comp__simps_I2_J,axiom,
! [B: $tType,C: $tType,A: $tType,M22: B > ( option @ A ),K2: B,K7: A,M1: A > ( option @ C )] :
( ( ( M22 @ K2 )
= ( some @ A @ K7 ) )
=> ( ( map_comp @ A @ C @ B @ M1 @ M22 @ K2 )
= ( M1 @ K7 ) ) ) ).
% map_comp_simps(2)
thf(fact_7642_map__comp__simps_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,M22: B > ( option @ A ),K2: B,M1: A > ( option @ C )] :
( ( ( M22 @ K2 )
= ( none @ A ) )
=> ( ( map_comp @ A @ C @ B @ M1 @ M22 @ K2 )
= ( none @ C ) ) ) ).
% map_comp_simps(1)
thf(fact_7643_map__comp__empty_I2_J,axiom,
! [B: $tType,D: $tType,C: $tType,M2: C > ( option @ B )] :
( ( map_comp @ B @ D @ C
@ ^ [X4: B] : ( none @ D )
@ M2 )
= ( ^ [X4: C] : ( none @ D ) ) ) ).
% map_comp_empty(2)
thf(fact_7644_map__comp__empty_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,M2: C > ( option @ B )] :
( ( map_comp @ C @ B @ A @ M2
@ ^ [X4: A] : ( none @ C ) )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% map_comp_empty(1)
thf(fact_7645_map__comp__Some__iff,axiom,
! [A: $tType,C: $tType,B: $tType,M1: B > ( option @ A ),M22: C > ( option @ B ),K2: C,V2: A] :
( ( ( map_comp @ B @ A @ C @ M1 @ M22 @ K2 )
= ( some @ A @ V2 ) )
= ( ? [K9: B] :
( ( ( M22 @ K2 )
= ( some @ B @ K9 ) )
& ( ( M1 @ K9 )
= ( some @ A @ V2 ) ) ) ) ) ).
% map_comp_Some_iff
thf(fact_7646_is__arg__min__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ( ( lattic501386751177426532rg_min @ A @ B )
= ( ^ [F4: A > B,P4: A > $o,X4: A] :
( ( P4 @ X4 )
& ! [Y3: A] :
( ( P4 @ Y3 )
=> ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( F4 @ Y3 ) ) ) ) ) ) ) ).
% is_arg_min_linorder
thf(fact_7647_is__arg__min__antimono,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F3: A > B,P: A > $o,X3: A,Y: A] :
( ( lattic501386751177426532rg_min @ A @ B @ F3 @ P @ X3 )
=> ( ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X3 ) )
=> ( ( P @ Y )
=> ( lattic501386751177426532rg_min @ A @ B @ F3 @ P @ Y ) ) ) ) ) ).
% is_arg_min_antimono
thf(fact_7648_map__comp__None__iff,axiom,
! [A: $tType,C: $tType,B: $tType,M1: B > ( option @ A ),M22: C > ( option @ B ),K2: C] :
( ( ( map_comp @ B @ A @ C @ M1 @ M22 @ K2 )
= ( none @ A ) )
= ( ( ( M22 @ K2 )
= ( none @ B ) )
| ? [K9: B] :
( ( ( M22 @ K2 )
= ( some @ B @ K9 ) )
& ( ( M1 @ K9 )
= ( none @ A ) ) ) ) ) ).
% map_comp_None_iff
thf(fact_7649_compute__powr__real,axiom,
( powr_real
= ( ^ [B8: real,I4: real] :
( if @ real @ ( ord_less_eq @ real @ B8 @ ( 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 @ B8 @ 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 @ B8 @ ( nat2 @ ( archim6421214686448440834_floor @ real @ I4 ) ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ B8 @ ( 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 @ B8 @ I4 ) ) ) ) ) ) ).
% compute_powr_real
thf(fact_7650_gen__length__def,axiom,
! [A: $tType] :
( ( gen_length @ A )
= ( ^ [N3: nat,Xs: list @ A] : ( plus_plus @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% gen_length_def
thf(fact_7651_gen__length__code_I2_J,axiom,
! [B: $tType,N: nat,X3: B,Xs2: list @ B] :
( ( gen_length @ B @ N @ ( cons @ B @ X3 @ Xs2 ) )
= ( gen_length @ B @ ( suc @ N ) @ Xs2 ) ) ).
% gen_length_code(2)
thf(fact_7652_length__code,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% length_code
thf(fact_7653_card__of__UNION__ordLeq__infinite,axiom,
! [B: $tType,A: $tType,C: $tType,B5: set @ A,I5: set @ B,A6: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B5 )
=> ( ( 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 @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ 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 @ ( A6 @ X5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B5 ) ) @ ( 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 ) @ A6 @ I5 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite
thf(fact_7654_inverse__rat__def,axiom,
( ( inverse_inverse @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X4: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X4 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_fst @ int @ int @ X4 ) ) ) ) ) ).
% inverse_rat_def
thf(fact_7655_card__of__ordLeqI,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,B5: set @ B] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ( member @ B @ ( F3 @ A5 ) @ B5 ) )
=> ( 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeqI
thf(fact_7656_card__of__image,axiom,
! [B: $tType,A: $tType,F3: B > A,A6: 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 @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_image
thf(fact_7657_card__of__UNION__Sigma,axiom,
! [B: $tType,A: $tType,A6: B > ( set @ A ),I5: 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 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I5 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A ) @ ( product_Sigma @ B @ A @ I5 @ A6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ).
% card_of_UNION_Sigma
thf(fact_7658_card__of__bool,axiom,
! [A: $tType,A1: A,A22: A] :
( ( A1 != A22 )
=> ( 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 @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ $o @ A ) ) ) ).
% card_of_bool
thf(fact_7659_card__of__empty,axiom,
! [B: $tType,A: $tType,A6: 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 @ A6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_empty
thf(fact_7660_card__of__empty2,axiom,
! [B: $tType,A: $tType,A6: 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty2
thf(fact_7661_card__of__empty3,axiom,
! [B: $tType,A: $tType,A6: 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty3
thf(fact_7662_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_7663_card__of__Times__infinite,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: set @ B] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( B5
!= ( 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 @ B5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( 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 @ A6
@ ^ [Uu3: A] : B5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) )
@ ( 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 @ B5
@ ^ [Uu3: B] : A6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ).
% card_of_Times_infinite
thf(fact_7664_card__of__Times__infinite__simps_I1_J,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( B5
!= ( 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 @ B5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( 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 @ A6
@ ^ [Uu3: A] : B5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(1)
thf(fact_7665_card__of__Times__infinite__simps_I2_J,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( B5
!= ( 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 @ B5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( 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 @ A6 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(2)
thf(fact_7666_card__of__Times__infinite__simps_I3_J,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: set @ B] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( B5
!= ( 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 @ B5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( 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 @ B5
@ ^ [Uu3: B] : A6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(3)
thf(fact_7667_card__of__Times__infinite__simps_I4_J,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( B5
!= ( 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 @ B5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( 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 @ A6 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B5
@ ^ [Uu3: B] : A6 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ B @ A ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(4)
thf(fact_7668_card__of__Times2,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: set @ B] :
( ( A6
!= ( 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 @ B5 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ).
% card_of_Times2
thf(fact_7669_card__of__Times1,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: set @ B] :
( ( A6
!= ( 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 @ B5 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B5
@ ^ [Uu3: B] : A6 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ B @ A ) ) ) ) ).
% card_of_Times1
thf(fact_7670_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: 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 ) ) @ R2 @ R4 ) @ ( bNF_We4044943003108391690rdLess @ 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 @ ( field2 @ A @ R2 ) ) @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ).
% BNF_Cardinal_Order_Relation.ordLess_Field
thf(fact_7671_card__of__Pow,axiom,
! [A: $tType,A6: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow2 @ A @ A6 ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ).
% card_of_Pow
thf(fact_7672_card__of__ordLeq__infinite,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ~ ( finite_finite2 @ A @ A6 )
=> ~ ( finite_finite2 @ B @ B5 ) ) ) ).
% card_of_ordLeq_infinite
thf(fact_7673_card__of__ordLeq__finite,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( finite_finite2 @ B @ B5 )
=> ( finite_finite2 @ A @ A6 ) ) ) ).
% card_of_ordLeq_finite
thf(fact_7674_card__of__ordIso__finite,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite2 @ A @ A6 )
= ( finite_finite2 @ B @ B5 ) ) ) ).
% card_of_ordIso_finite
thf(fact_7675_infinite__iff__card__of__nat,axiom,
! [A: $tType,A6: set @ A] :
( ( ~ ( finite_finite2 @ A @ A6 ) )
= ( 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 @ A6 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_card_of_nat
thf(fact_7676_card__of__Sigma__ordLeq__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,B5: set @ A,I5: set @ B,A6: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B5 )
=> ( ( 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 @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ 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 @ ( A6 @ X5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B5 ) ) @ ( 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 @ A6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite
thf(fact_7677_card__of__Times__same__infinite,axiom,
! [A: $tType,A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( 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 @ A6
@ ^ [Uu3: A] : A6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% card_of_Times_same_infinite
thf(fact_7678_card__of__Times3,axiom,
! [A: $tType,A6: set @ A] :
( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu3: A] : A6 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ A ) ) ) ).
% card_of_Times3
thf(fact_7679_card__of__Sigma__mono1,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,A6: A > ( set @ B ),B5: A > ( set @ C )] :
( ! [X5: A] :
( ( member @ A @ X5 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( A6 @ X5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( B5 @ X5 ) ) ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ I5 @ A6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C ) @ ( product_Sigma @ A @ C @ I5 @ B5 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Sigma_mono1
thf(fact_7680_card__of__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A6: set @ A,B5: set @ B,C4: set @ 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C )
@ ( product_Sigma @ A @ C @ A6
@ ^ [Uu3: A] : C4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ B5
@ ^ [Uu3: B] : C4 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% card_of_Times_mono1
thf(fact_7681_card__of__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A6: set @ A,B5: set @ B,C4: set @ 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ A )
@ ( product_Sigma @ C @ A @ C4
@ ^ [Uu3: C] : A6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
@ ( product_Sigma @ C @ B @ C4
@ ^ [Uu3: C] : B5 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% card_of_Times_mono2
thf(fact_7682_card__of__Times__commute,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B5
@ ^ [Uu3: B] : A6 ) ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% card_of_Times_commute
thf(fact_7683_card__of__Func__Times,axiom,
! [C: $tType,B: $tType,A: $tType,A6: set @ A,B5: set @ B,C4: set @ C] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ A @ B ) > C ) ) ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( A > B > C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ A @ B ) > C ) ) ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( A > B > C ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( ( product_prod @ A @ B ) > C )
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 )
@ C4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B > C ) @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A6 @ ( bNF_Wellorder_Func @ B @ C @ B5 @ C4 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( ( product_prod @ A @ B ) > C ) @ ( A > B > C ) ) ) ).
% card_of_Func_Times
thf(fact_7684_Func__Times__Range,axiom,
! [C: $tType,B: $tType,A: $tType,A6: set @ A,B5: set @ B,C4: set @ C] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > ( product_prod @ B @ C ) )
@ ( bNF_Wellorder_Func @ A @ ( product_prod @ B @ C ) @ A6
@ ( product_Sigma @ B @ C @ B5
@ ^ [Uu3: B] : C4 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ ( A > B ) @ ( A > C ) )
@ ( product_Sigma @ ( A > B ) @ ( A > C ) @ ( bNF_Wellorder_Func @ A @ B @ A6 @ B5 )
@ ^ [Uu3: A > B] : ( bNF_Wellorder_Func @ A @ C @ A6 @ C4 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% Func_Times_Range
thf(fact_7685_card__of__mono2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: 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 ) ) @ R2 @ R4 ) @ ( 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 @ ( field2 @ A @ R2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_mono2
thf(fact_7686_card__of__cong,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordIso @ 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 @ ( field2 @ A @ R2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R4 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_cong
thf(fact_7687_card__of__Pow__Func,axiom,
! [A: $tType,A6: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow2 @ A @ A6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( A > $o ) @ ( bNF_Wellorder_Func @ A @ $o @ A6 @ ( top_top @ ( set @ $o ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( set @ A ) @ ( A > $o ) ) ) ).
% card_of_Pow_Func
thf(fact_7688_card__of__refl,axiom,
! [A: $tType,A6: 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ).
% card_of_refl
thf(fact_7689_card__of__mono1,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_mono1
thf(fact_7690_card__of__least,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A6 @ 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_least
thf(fact_7691_type__copy__set__bd,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType,S3: A > ( set @ B ),Bd: set @ ( product_prod @ C @ C ),Rep: D > A,X3: D] :
( ! [Y4: A] : ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( S3 @ Y4 ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( comp @ A @ ( set @ B ) @ D @ S3 @ Rep @ X3 ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) ) ).
% type_copy_set_bd
thf(fact_7692_ex__bij__betw,axiom,
! [B: $tType,A: $tType,A6: 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 @ A6 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ? [F2: B > A,B7: set @ B] : ( bij_betw @ B @ A @ F2 @ B7 @ A6 ) ) ).
% ex_bij_betw
thf(fact_7693_card__of__ordIsoI,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,B5: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A6 @ B5 )
=> ( 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_ordIsoI
thf(fact_7694_card__of__ordIso,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] :
( ( ? [F4: A > B] : ( bij_betw @ A @ B @ F4 @ A6 @ B5 ) )
= ( 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_ordIso
thf(fact_7695_infinite__iff__natLeq__ordLeq,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
!= ( 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 @ A6 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_natLeq_ordLeq
thf(fact_7696_finite__iff__ordLess__natLeq,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A7: 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 @ A7 ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ A @ nat ) ) ) ) ).
% finite_iff_ordLess_natLeq
thf(fact_7697_card__of__nat,axiom,
member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( top_top @ ( set @ nat ) ) ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_Wellorder_ordIso @ nat @ nat ) ).
% card_of_nat
thf(fact_7698_card__of__Field__natLeq,axiom,
member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( field2 @ nat @ bNF_Ca8665028551170535155natLeq ) ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_Wellorder_ordIso @ nat @ nat ) ).
% card_of_Field_natLeq
thf(fact_7699_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [G4: B > A] :
( ( image2 @ B @ A @ G4 @ B5 )
= A6 ) )
= ( 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeq2
thf(fact_7700_surj__imp__ordLeq,axiom,
! [B: $tType,A: $tType,B5: set @ A,F3: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( 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 @ B5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% surj_imp_ordLeq
thf(fact_7701_card__of__singl__ordLeq,axiom,
! [A: $tType,B: $tType,A6: set @ A,B2: B] :
( ( A6
!= ( 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 @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_singl_ordLeq
thf(fact_7702_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B5: set @ A,A6: set @ B] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ? [F4: B > A] :
( ( image2 @ B @ A @ F4 @ A6 )
= B5 ) )
= ( 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B5 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ) ).
% card_of_ordLess2
thf(fact_7703_internalize__card__of__ordLeq2,axiom,
! [A: $tType,B: $tType,A6: set @ A,C4: 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ C4 )
& ( 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 @ A6 ) @ ( 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 @ C4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq2
thf(fact_7704_card__of__Field__ordLess,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R2 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_Field_ordLess
thf(fact_7705_card__of__Func__UNIV,axiom,
! [B: $tType,A: $tType,B5: 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 ) ) @ B5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F4: A > B] : ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) ) @ B5 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > B ) @ ( A > B ) ) ) ).
% card_of_Func_UNIV
thf(fact_7706_ordLeq__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),C4: set @ 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C )
@ ( product_Sigma @ A @ C @ ( field2 @ A @ R2 )
@ ^ [Uu3: A] : C4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ ( field2 @ B @ R4 )
@ ^ [Uu3: B] : C4 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% ordLeq_Times_mono1
thf(fact_7707_ordLeq__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),A6: set @ 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ A )
@ ( product_Sigma @ C @ A @ A6
@ ^ [Uu3: C] : ( field2 @ A @ R2 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
@ ( product_Sigma @ C @ B @ A6
@ ^ [Uu3: C] : ( field2 @ B @ R4 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% ordLeq_Times_mono2
thf(fact_7708_uminus__rat__def,axiom,
( ( uminus_uminus @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X4 ) ) @ ( product_snd @ int @ int @ X4 ) ) ) ) ).
% uminus_rat_def
thf(fact_7709_card__of__ordLeq,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] :
( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A6 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A6 ) @ B5 ) ) )
= ( 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_ordLeq
thf(fact_7710_internalize__card__of__ordLeq,axiom,
! [A: $tType,B: $tType,A6: 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 @ A6 ) @ 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 @ A6 ) @ ( 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_7711_card__of__ordLess,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: set @ B] :
( ( ~ ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A6 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A6 ) @ B5 ) ) )
= ( 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 @ B5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ).
% card_of_ordLess
thf(fact_7712_regularCard__def,axiom,
! [A: $tType] :
( ( bNF_Ca7133664381575040944arCard @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [K6: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ K6 @ ( field2 @ A @ R5 ) )
& ( bNF_Ca7293521722713021262ofinal @ A @ K6 @ 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 @ K6 ) @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% regularCard_def
thf(fact_7713_ordLeq3__finite__infinite,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ~ ( finite_finite2 @ B @ B5 )
=> ( 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% ordLeq3_finite_infinite
thf(fact_7714_comp__set__bd__Union__o__collect,axiom,
! [C: $tType,B: $tType,A: $tType,X3: C,X6: set @ ( C > ( set @ ( set @ A ) ) ),Hbd: 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
@ ( complete_Sup_Sup @ ( set @ A )
@ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ ( C > ( set @ ( set @ A ) ) ) @ ( set @ ( set @ A ) )
@ ^ [F4: C > ( set @ ( set @ A ) )] : ( F4 @ X3 )
@ X6 ) ) ) )
@ Hbd )
@ ( 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 @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ C @ ( complete_Sup_Sup @ ( set @ A ) ) @ ( bNF_collect @ C @ ( set @ A ) @ X6 ) @ X3 ) ) @ Hbd ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% comp_set_bd_Union_o_collect
thf(fact_7715_plus__rat__def,axiom,
( ( plus_plus @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ rep_Rat @ ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat )
@ ^ [X4: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ) ).
% plus_rat_def
thf(fact_7716_times__rat__def,axiom,
( ( times_times @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ rep_Rat @ ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat )
@ ^ [X4: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ) ).
% times_rat_def
thf(fact_7717_card__of__ordIso__subst,axiom,
! [A: $tType,A6: set @ A,B5: set @ A] :
( ( A6 = B5 )
=> ( 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B5 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_ordIso_subst
thf(fact_7718_card__of__Sigma__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),I5: set @ B,A6: 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 ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ 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 @ ( A6 @ X5 ) ) @ 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 @ A6 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite_Field
thf(fact_7719_dir__image,axiom,
! [B: $tType,A: $tType,F3: A > B,R2: set @ ( product_prod @ A @ A )] :
( ! [X5: A,Y4: A] :
( ( ( F3 @ X5 )
= ( F3 @ Y4 ) )
= ( X5 = Y4 ) )
=> ( ( 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_We2720479622203943262_image @ A @ B @ R2 @ F3 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% dir_image
thf(fact_7720_Card__order__trans,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A,Z2: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( X3 != Y )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( ( Y != Z2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R2 )
=> ( ( X3 != Z2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ R2 ) ) ) ) ) ) ) ).
% Card_order_trans
thf(fact_7721_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 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R2 ) )
& ( A3 != X5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X5 ) @ R2 ) ) ) ) ) ).
% infinite_Card_order_limit
thf(fact_7722_Card__order__ordIso2,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ 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 @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 ) ) ) ).
% Card_order_ordIso2
thf(fact_7723_Card__order__ordIso,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( 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 ) ) @ R4 @ R2 ) @ ( bNF_Wellorder_ordIso @ B @ A ) )
=> ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 ) ) ) ).
% Card_order_ordIso
thf(fact_7724_ordLeq__refl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ 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 ) ) @ R2 @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% ordLeq_refl
thf(fact_7725_ordIso__refl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ 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 ) ) @ R2 @ R2 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% ordIso_refl
thf(fact_7726_card__order__on__ordIso,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A6 @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ A6 @ 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 ) ) ) ) ).
% card_order_on_ordIso
thf(fact_7727_card__order__on__def,axiom,
! [A: $tType] :
( ( bNF_Ca8970107618336181345der_on @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A7 @ R5 )
& ! [R9: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A7 @ R9 )
=> ( 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 ) ) @ R5 @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% card_order_on_def
thf(fact_7728_card__of__unique,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A6 @ 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 ) ) @ R2 @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_unique
thf(fact_7729_Card__order__iff__ordLeq__card__of,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ 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 ) ) @ R2 @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R2 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% Card_order_iff_ordLeq_card_of
thf(fact_7730_card__of__Field__ordIso,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R2 ) ) @ R2 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_Field_ordIso
thf(fact_7731_Card__order__iff__ordIso__card__of,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ 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 ) ) @ R2 @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R2 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% Card_order_iff_ordIso_card_of
thf(fact_7732_ordIso__card__of__imp__Card__order,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),A6: 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 ) ) @ R2 @ ( bNF_Ca6860139660246222851ard_of @ B @ A6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) ) ).
% ordIso_card_of_imp_Card_order
thf(fact_7733_exists__minim__Card__order,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X5: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X5 @ R )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ X5 ) @ X5 ) )
=> ? [X5: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X5 @ R )
& ! [Xa: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa @ R )
=> ( 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 ) ) @ X5 @ Xa ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Card_order
thf(fact_7734_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_7735_card__of__ordIso__finite__Field,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),A6: 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 @ A6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
= ( finite_finite2 @ B @ A6 ) ) ) ) ).
% card_of_ordIso_finite_Field
thf(fact_7736_card__of__underS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A3 @ ( 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( order_underS @ A @ R2 @ A3 ) ) @ R2 ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% card_of_underS
thf(fact_7737_Card__order__singl__ordLeq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),B2: 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 @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% Card_order_singl_ordLeq
thf(fact_7738_card__of__Un__ordLeq__infinite__Field,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ B,B5: 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 @ A6 ) @ 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 @ B5 ) @ 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 ) @ A6 @ B5 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ) ).
% card_of_Un_ordLeq_infinite_Field
thf(fact_7739_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_7740_Card__order__Pow,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R2 @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow2 @ A @ ( field2 @ A @ R2 ) ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% Card_order_Pow
thf(fact_7741_Card__order__Times2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( A6
!= ( 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 @ A6
@ ^ [Uu3: B] : ( field2 @ A @ R2 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ) ) ).
% Card_order_Times2
thf(fact_7742_Card__order__Times1,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( B5
!= ( 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] : B5 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ B ) ) ) ) ) ).
% Card_order_Times1
thf(fact_7743_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_7744_card__of__UNION__ordLeq__infinite__Field,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set @ ( product_prod @ A @ A ),I5: set @ B,A6: 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 ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ 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 @ ( A6 @ X5 ) ) @ 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 ) @ A6 @ I5 ) ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite_Field
thf(fact_7745_Card__order__iff__Restr__underS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( 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 @ ( order_underS @ A @ R2 @ X4 )
@ ^ [Uu3: A] : ( order_underS @ A @ R2 @ X4 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R2 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% Card_order_iff_Restr_underS
thf(fact_7746_regularCard__UNION,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),As4: A > ( set @ B ),B5: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca7133664381575040944arCard @ A @ R2 )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ ( set @ B ) @ R2 @ As4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ As4 @ ( 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 @ B5 ) @ R2 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R2 ) )
& ( ord_less_eq @ ( set @ B ) @ B5 @ ( As4 @ X5 ) ) ) ) ) ) ) ) ).
% regularCard_UNION
thf(fact_7747_Card__order__Times__infinite,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),P2: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ( field2 @ B @ P2 )
!= ( 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 ) ) @ P2 @ 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 @ P2 ) ) )
@ 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 @ P2 )
@ ^ [Uu3: B] : ( field2 @ A @ R2 ) ) )
@ R2 )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ) ).
% Card_order_Times_infinite
thf(fact_7748_card__of__Times__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ B,B5: 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 @ A6 ) @ 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 @ B5 ) @ 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 @ A6
@ ^ [Uu3: B] : B5 ) )
@ R2 )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Times_ordLeq_infinite_Field
thf(fact_7749_ex__toCard__pred,axiom,
! [B: $tType,A: $tType,A6: 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 @ A6 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ? [X_12: A > B] : ( bNF_Gr1419584066657907630d_pred @ A @ B @ A6 @ R2 @ X_12 ) ) ) ).
% ex_toCard_pred
thf(fact_7750_toCard__pred__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr1419584066657907630d_pred @ A @ B )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ B @ B ),F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A7 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A7 ) @ ( field2 @ B @ R5 ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 ) ) ) ) ).
% toCard_pred_def
thf(fact_7751_toCard__pred__toCard,axiom,
! [A: $tType,B: $tType,A6: 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 @ A6 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ( bNF_Gr1419584066657907630d_pred @ A @ B @ A6 @ R2 @ ( bNF_Greatest_toCard @ A @ B @ A6 @ R2 ) ) ) ) ).
% toCard_pred_toCard
thf(fact_7752_cardSuc__UNION,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),As4: ( set @ A ) > ( set @ B ),B5: 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 ) @ As4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As4 @ ( 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 @ B5 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) )
& ( ord_less_eq @ ( set @ B ) @ B5 @ ( As4 @ X5 ) ) ) ) ) ) ) ) ).
% cardSuc_UNION
thf(fact_7753_cardSuc__ordLeq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R2 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( set @ A ) ) ) ) ).
% cardSuc_ordLeq
thf(fact_7754_cardSuc__greater,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R2 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% cardSuc_greater
thf(fact_7755_cardSuc__least,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ 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 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) @ R4 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% cardSuc_least
thf(fact_7756_cardSuc__ordLess__ordLeq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ 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 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) @ R4 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% cardSuc_ordLess_ordLeq
thf(fact_7757_cardSuc__mono__ordLeq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) @ ( bNF_Ca8387033319878233205ardSuc @ B @ R4 ) ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% cardSuc_mono_ordLeq
thf(fact_7758_cardSuc__invar__ordIso,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) @ ( bNF_Ca8387033319878233205ardSuc @ B @ R4 ) ) @ ( bNF_Wellorder_ordIso @ ( set @ A ) @ ( 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% cardSuc_invar_ordIso
thf(fact_7759_cardSuc__least__aux,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R4 ) @ R4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R2 @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) @ R4 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ).
% cardSuc_least_aux
thf(fact_7760_cardSuc__ordLeq__ordLess,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R4 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ ( 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 ) ) @ R4 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% cardSuc_ordLeq_ordLess
thf(fact_7761_toCard__inj,axiom,
! [B: $tType,A: $tType,A6: set @ A,R2: set @ ( product_prod @ B @ B ),X3: A,Y: 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 @ A6 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( member @ A @ Y @ A6 )
=> ( ( ( bNF_Greatest_toCard @ A @ B @ A6 @ R2 @ X3 )
= ( bNF_Greatest_toCard @ A @ B @ A6 @ R2 @ Y ) )
= ( X3 = Y ) ) ) ) ) ) ).
% toCard_inj
thf(fact_7762_fromCard__toCard,axiom,
! [B: $tType,A: $tType,A6: set @ A,R2: set @ ( product_prod @ B @ B ),B2: 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 @ A6 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ( ( member @ A @ B2 @ A6 )
=> ( ( bNF_Gr5436034075474128252omCard @ A @ B @ A6 @ R2 @ ( bNF_Greatest_toCard @ A @ B @ A6 @ R2 @ B2 ) )
= B2 ) ) ) ) ).
% fromCard_toCard
thf(fact_7763_isCardSuc__def,axiom,
! [A: $tType] :
( ( bNF_Ca6246979054910435723ardSuc @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R9 ) @ R9 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R5 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
& ! [R10: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R10 ) @ R10 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R5 @ R10 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R9 @ R10 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) ).
% isCardSuc_def
thf(fact_7764_cardSuc__UNION__Cinfinite,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),As4: ( set @ A ) > ( set @ B ),B5: set @ B] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) @ As4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As4 @ ( 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 @ B5 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) )
& ( ord_less_eq @ ( set @ B ) @ B5 @ ( As4 @ X5 ) ) ) ) ) ) ) ).
% cardSuc_UNION_Cinfinite
thf(fact_7765_comp__single__set__bd,axiom,
! [B: $tType,D: $tType,A: $tType,E: $tType,C: $tType,Fbd: set @ ( product_prod @ A @ A ),Fset: B > ( set @ C ),Gset: D > ( set @ B ),Gbd: set @ ( product_prod @ E @ E ),X3: D] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ Fbd ) @ Fbd )
=> ( ! [X5: B] : ( 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 @ ( Fset @ X5 ) ) @ Fbd ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ! [X5: D] : ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( Gset @ X5 ) ) @ Gbd ) @ ( bNF_Wellorder_ordLeq @ B @ E ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ ( product_prod @ E @ A ) @ ( product_prod @ E @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ ( product_prod @ E @ A ) @ ( product_prod @ E @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ Fset @ ( Gset @ X3 ) ) ) ) @ ( bNF_Cardinal_cprod @ E @ A @ Gbd @ Fbd ) ) @ ( bNF_Wellorder_ordLeq @ C @ ( product_prod @ E @ A ) ) ) ) ) ) ).
% comp_single_set_bd
thf(fact_7766_cprod__infinite,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ 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_Cardinal_cprod @ A @ A @ R2 @ R2 ) @ R2 ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% cprod_infinite
thf(fact_7767_cinfinite__mono,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: 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 ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca4139267488887388095finite @ A @ R1 )
=> ( bNF_Ca4139267488887388095finite @ B @ R22 ) ) ) ).
% cinfinite_mono
thf(fact_7768_cprod__com,axiom,
! [B: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),P22: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Cardinal_cprod @ A @ B @ P1 @ P22 ) @ ( bNF_Cardinal_cprod @ B @ A @ P22 @ P1 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% cprod_com
thf(fact_7769_Cinfinite__limit2,axiom,
! [A: $tType,X1: A,R2: set @ ( product_prod @ A @ A ),X2: A] :
( ( member @ A @ X1 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ X2 @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R2 ) )
& ( X1 != X5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X1 @ X5 ) @ R2 )
& ( X2 != X5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X5 ) @ R2 ) ) ) ) ) ).
% Cinfinite_limit2
thf(fact_7770_Cinfinite__limit,axiom,
! [A: $tType,X3: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R2 ) )
& ( X3 != X5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X5 ) @ R2 ) ) ) ) ).
% Cinfinite_limit
thf(fact_7771_cprod__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P2: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),Q3: 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 ) ) @ P2 @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q3 @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P2 ) @ P2 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P2 @ Q3 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ B ) ) ) ) ) ) ) ).
% cprod_cinfinite_bound
thf(fact_7772_cprod__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),P2: set @ ( product_prod @ B @ B ),P8: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca4139267488887388095finite @ A @ R2 )
=> ( ( 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 @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( bNF_Cardinal_cprod @ B @ C @ P2 @ P8 ) @ ( bNF_Cardinal_cprod @ A @ A @ R2 @ R2 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ C ) @ ( product_prod @ A @ 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_Cardinal_cprod @ B @ C @ P2 @ P8 ) @ R2 ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% cprod_dup
thf(fact_7773_Cinfinite__cong,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: 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 ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R1 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
=> ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) ) ) ) ).
% Cinfinite_cong
thf(fact_7774_Cinfinite__limit__finite,axiom,
! [A: $tType,X6: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R2 ) )
& ! [Xa: A] :
( ( member @ A @ Xa @ X6 )
=> ( ( Xa != X5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa @ X5 ) @ R2 ) ) ) ) ) ) ) ).
% Cinfinite_limit_finite
thf(fact_7775_cprod__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: 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 ) ) @ P1 @ R1 ) @ ( 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 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_cprod @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% cprod_mono
thf(fact_7776_cprod__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: 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 ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P1 @ Q3 ) @ ( bNF_Cardinal_cprod @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% cprod_mono1
thf(fact_7777_cprod__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: 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 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) @ ( bNF_Cardinal_cprod @ C @ A @ Q3 @ P22 ) @ ( bNF_Cardinal_cprod @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% cprod_mono2
thf(fact_7778_cprod__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: 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 ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordIso @ 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 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_cprod @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% cprod_cong
thf(fact_7779_cprod__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: 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 ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_cprod @ B @ C @ R1 @ P22 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% cprod_cong1
thf(fact_7780_cprod__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: 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 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) @ ( bNF_Cardinal_cprod @ C @ A @ Q3 @ P22 ) @ ( bNF_Cardinal_cprod @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% cprod_cong2
thf(fact_7781_Un__Cinfinite__bound,axiom,
! [B: $tType,A: $tType,A6: set @ A,R2: set @ ( product_prod @ B @ B ),B5: 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 @ A6 ) @ 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 @ B5 ) @ 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 ) @ A6 @ B5 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% Un_Cinfinite_bound
thf(fact_7782_natLeq__ordLeq__cinfinite,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( 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 @ R2 ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% natLeq_ordLeq_cinfinite
thf(fact_7783_UNION__Cinfinite__bound,axiom,
! [A: $tType,B: $tType,C: $tType,I5: set @ A,R2: set @ ( product_prod @ B @ B ),A6: A > ( set @ 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ I5 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A6 @ X5 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ A6 @ I5 ) ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) ) ) ) ).
% UNION_Cinfinite_bound
thf(fact_7784_card__of__Csum__Times_H,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),I5: set @ B,A6: B > ( set @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ 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 @ ( A6 @ X5 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) )
@ ( bNF_Cardinal_Csum @ B @ C @ ( bNF_Ca6860139660246222851ard_of @ B @ I5 )
@ ^ [I4: B] : ( bNF_Ca6860139660246222851ard_of @ C @ ( A6 @ I4 ) ) )
@ ( bNF_Cardinal_cprod @ B @ A @ ( bNF_Ca6860139660246222851ard_of @ B @ I5 ) @ R2 ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ ( product_prod @ B @ A ) ) ) ) ) ).
% card_of_Csum_Times'
thf(fact_7785_card__of__Csum__Times,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,A6: A > ( set @ B ),B5: set @ C] :
( ! [X5: A] :
( ( member @ A @ X5 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( A6 @ X5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) )
@ ( bNF_Cardinal_Csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ I5 )
@ ^ [I4: A] : ( bNF_Ca6860139660246222851ard_of @ B @ ( A6 @ I4 ) ) )
@ ( bNF_Cardinal_cprod @ A @ C @ ( bNF_Ca6860139660246222851ard_of @ A @ I5 ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B5 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Csum_Times
thf(fact_7786_Cfinite__cprod__Cinfinite,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ S )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S ) @ S ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R2 @ S ) @ S ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ B ) ) ) ) ).
% Cfinite_cprod_Cinfinite
thf(fact_7787_cprod__infinite1_H,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P2: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( ( ~ ( 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 ) ) @ P2 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ P2 ) @ P2 ) )
=> ( ( 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 ) ) @ P2 @ 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_Cardinal_cprod @ A @ B @ R2 @ P2 ) @ R2 ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% cprod_infinite1'
thf(fact_7788_czero__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_Cardinal_czero @ A ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ).
% czero_ordIso
thf(fact_7789_czero__def,axiom,
! [A: $tType] :
( ( bNF_Cardinal_czero @ A )
= ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% czero_def
thf(fact_7790_cinfinite__not__czero,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca4139267488887388095finite @ B @ 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 ) ) @ R2 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ).
% cinfinite_not_czero
thf(fact_7791_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_7792_card__of__ordIso__czero__iff__empty,axiom,
! [B: $tType,A: $tType,A6: 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 @ A6 ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_ordIso_czero_iff_empty
thf(fact_7793_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_7794_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_7795_Cnotzero__mono,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),Q3: set @ ( product_prod @ B @ B )] :
( ( ~ ( 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 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q3 ) @ Q3 )
=> ( ( 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 @ Q3 ) @ ( bNF_Wellorder_ordLeq @ 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 ) ) @ Q3 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q3 ) @ Q3 ) ) ) ) ) ).
% Cnotzero_mono
thf(fact_7796_Cinfinite__Cnotzero,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ 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 ) ) @ R2 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) ) ) ).
% Cinfinite_Cnotzero
thf(fact_7797_Cnotzero__UNIV,axiom,
! [A: $tType] :
( ~ ( 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 @ ( top_top @ ( set @ A ) ) ) @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% Cnotzero_UNIV
thf(fact_7798_Cinfinite__cprod2,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( ~ ( 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 ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
=> ( ( bNF_Ca4139267488887388095finite @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) )
& ( bNF_Ca8970107618336181345der_on @ ( product_prod @ A @ B ) @ ( field2 @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) ) ) ) ).
% Cinfinite_cprod2
thf(fact_7799_cinfinite__cprod2,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( ~ ( 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 ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
=> ( bNF_Ca4139267488887388095finite @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) ) ) ).
% cinfinite_cprod2
thf(fact_7800_ordLeq__cprod2,axiom,
! [A: $tType,B: $tType,P1: set @ ( product_prod @ A @ A ),P22: set @ ( product_prod @ B @ B )] :
( ( ~ ( 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 ) ) @ P1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P1 ) @ P1 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ P22 ) @ P22 )
=> ( 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 ) ) ) @ P22 @ ( bNF_Cardinal_cprod @ A @ B @ P1 @ P22 ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ) ).
% ordLeq_cprod2
thf(fact_7801_Cfinite__ordLess__Cinfinite,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ S )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S ) @ S ) )
=> ( 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 @ S ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ).
% Cfinite_ordLess_Cinfinite
thf(fact_7802_cone__ordLeq__Cnotzero,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 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_cone @ R2 ) @ ( bNF_Wellorder_ordLeq @ product_unit @ A ) ) ) ).
% cone_ordLeq_Cnotzero
thf(fact_7803_cexp__mono2__Cnotzero,axiom,
! [B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: 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 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( ( ~ ( 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 ) ) @ P22 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P22 ) @ P22 ) )
=> ( 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 @ Q3 @ P22 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2_Cnotzero
thf(fact_7804_cone__not__czero,axiom,
! [A: $tType] :
~ ( member @ ( product_prod @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_cone @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ product_unit @ A ) ) ).
% cone_not_czero
thf(fact_7805_cexp__cprod,axiom,
! [A: $tType,C: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ C @ C ),R32: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( B > C > A ) @ ( B > C > A ) ) ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ B ) > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( B > C > A ) @ ( B > C > A ) ) ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ B ) > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) @ ( bNF_Cardinal_cexp @ ( C > A ) @ B @ ( bNF_Cardinal_cexp @ A @ C @ R1 @ R22 ) @ R32 ) @ ( bNF_Cardinal_cexp @ A @ ( product_prod @ C @ B ) @ R1 @ ( bNF_Cardinal_cprod @ C @ B @ R22 @ R32 ) ) ) @ ( bNF_Wellorder_ordIso @ ( B > C > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) ) ).
% cexp_cprod
thf(fact_7806_cprod__cexp,axiom,
! [C: $tType,B: $tType,A: $tType,R2: set @ ( product_prod @ B @ B ),S: set @ ( product_prod @ C @ C ),T2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) @ ( bNF_Cardinal_cexp @ ( product_prod @ B @ C ) @ A @ ( bNF_Cardinal_cprod @ B @ C @ R2 @ S ) @ T2 ) @ ( bNF_Cardinal_cprod @ ( A > B ) @ ( A > C ) @ ( bNF_Cardinal_cexp @ B @ A @ R2 @ T2 ) @ ( bNF_Cardinal_cexp @ C @ A @ S @ T2 ) ) ) @ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% cprod_cexp
thf(fact_7807_cexp__cone,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_unit > A ) @ ( product_unit > A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_unit > A ) @ ( product_unit > A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cexp @ A @ product_unit @ R2 @ bNF_Cardinal_cone ) @ R2 ) @ ( bNF_Wellorder_ordIso @ ( product_unit > A ) @ A ) ) ) ).
% cexp_cone
thf(fact_7808_cexp__cprod__ordLeq,axiom,
! [A: $tType,B: $tType,C: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R32: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R32 @ ( bNF_Cardinal_czero @ C ) ) @ ( bNF_Wellorder_ordIso @ C @ C ) )
& ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ R32 ) @ R32 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R32 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > B > A ) @ ( C > B > A ) ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > B > A ) @ ( C > B > A ) ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( bNF_Cardinal_cexp @ ( B > A ) @ C @ ( bNF_Cardinal_cexp @ A @ B @ R1 @ R22 ) @ R32 ) @ ( bNF_Cardinal_cexp @ A @ B @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( C > B > A ) @ ( B > A ) ) ) ) ) ) ) ).
% cexp_cprod_ordLeq
thf(fact_7809_cexp__mono_H,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: 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 ) ) @ P1 @ R1 ) @ ( 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 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( ( ( ( field2 @ C @ P22 )
= ( bot_bot @ ( set @ C ) ) )
=> ( ( field2 @ D @ R22 )
= ( 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 @ P1 @ P22 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ).
% cexp_mono'
thf(fact_7810_cexp__mono1,axiom,
! [B: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: 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 ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P1 @ Q3 ) @ ( bNF_Cardinal_cexp @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( C > B ) ) ) ) ) ).
% cexp_mono1
thf(fact_7811_cexp__mono2_H,axiom,
! [B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: 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 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( ( ( ( field2 @ A @ P22 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( field2 @ B @ R22 )
= ( 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 @ Q3 @ P22 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2'
thf(fact_7812_ordLeq__cexp1,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),Q3: set @ ( product_prod @ B @ B )] :
( ( ~ ( 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 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q3 ) @ Q3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ Q3 @ ( bNF_Cardinal_cexp @ B @ A @ Q3 @ R2 ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( A > B ) ) ) ) ) ).
% ordLeq_cexp1
thf(fact_7813_cexp__cong,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: 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 ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordIso @ 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 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( ( bNF_Ca8970107618336181345der_on @ D @ ( field2 @ D @ R22 ) @ R22 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P22 ) @ P22 )
=> ( 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 @ P1 @ P22 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% cexp_cong
thf(fact_7814_cexp__cong1,axiom,
! [B: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: 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 ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P1 @ Q3 ) @ ( bNF_Cardinal_cexp @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( C > B ) ) ) ) ) ).
% cexp_cong1
thf(fact_7815_cexp__cong2,axiom,
! [B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: 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 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P22 ) @ P22 )
=> ( 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 @ Q3 @ P22 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_cong2
thf(fact_7816_cexp__mono,axiom,
! [E: $tType,F: $tType,B: $tType,D: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: 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 ) ) @ P1 @ R1 ) @ ( 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 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ E @ E ) ) @ P22 @ ( bNF_Cardinal_czero @ E ) ) @ ( bNF_Wellorder_ordIso @ C @ E ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ D @ D ) ) @ ( set @ ( product_prod @ F @ F ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ D @ D ) ) @ ( set @ ( product_prod @ F @ F ) ) @ R22 @ ( bNF_Cardinal_czero @ F ) ) @ ( bNF_Wellorder_ordIso @ D @ F ) ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P22 ) @ P22 )
=> ( 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 @ P1 @ P22 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% cexp_mono
thf(fact_7817_cexp__mono2,axiom,
! [D: $tType,E: $tType,B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: 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 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ ( bNF_Cardinal_czero @ D ) ) @ ( bNF_Wellorder_ordIso @ A @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) @ R22 @ ( bNF_Cardinal_czero @ E ) ) @ ( bNF_Wellorder_ordIso @ B @ E ) ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P22 ) @ P22 )
=> ( 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 @ Q3 @ P22 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ) ).
% cexp_mono2
thf(fact_7818_ordLeq__cexp2,axiom,
! [A: $tType,B: $tType,Q3: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B )] :
( ( 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_Cardinal_ctwo @ Q3 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ R2 @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R2 ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( B > A ) ) ) ) ) ).
% ordLeq_cexp2
thf(fact_7819_Cfinite__cexp__Cinfinite,axiom,
! [A: $tType,B: $tType,S: set @ ( product_prod @ A @ A ),T2: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ S )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ S ) @ S ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ T2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ T2 ) @ T2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( set @ ( product_prod @ ( B > $o ) @ ( B > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( set @ ( product_prod @ ( B > $o ) @ ( B > $o ) ) ) @ ( bNF_Cardinal_cexp @ A @ B @ S @ T2 ) @ ( bNF_Cardinal_cexp @ $o @ B @ bNF_Cardinal_ctwo @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( B > A ) @ ( B > $o ) ) ) ) ) ).
% Cfinite_cexp_Cinfinite
thf(fact_7820_ctwo__Cnotzero,axiom,
( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ $o @ $o ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ $o @ $o ) ) @ bNF_Cardinal_ctwo @ ( bNF_Cardinal_czero @ $o ) ) @ ( bNF_Wellorder_ordIso @ $o @ $o ) )
& ( bNF_Ca8970107618336181345der_on @ $o @ ( field2 @ $o @ bNF_Cardinal_ctwo ) @ bNF_Cardinal_ctwo ) ) ).
% ctwo_Cnotzero
thf(fact_7821_ctwo__ordLess__natLeq,axiom,
member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Cardinal_ctwo @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ $o @ nat ) ).
% ctwo_ordLess_natLeq
thf(fact_7822_ordLess__ctwo__cexp,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) @ R2 @ ( bNF_Cardinal_cexp @ $o @ A @ bNF_Cardinal_ctwo @ R2 ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( A > $o ) ) ) ) ).
% ordLess_ctwo_cexp
thf(fact_7823_ctwo__not__czero,axiom,
! [A: $tType] :
~ ( 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_Cardinal_ctwo @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ $o @ A ) ) ).
% ctwo_not_czero
thf(fact_7824_ctwo__ordLeq__Cinfinite,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( 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_Cardinal_ctwo @ R2 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) ) ) ).
% ctwo_ordLeq_Cinfinite
thf(fact_7825_ctwo__ordLess__Cinfinite,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( 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_Cardinal_ctwo @ R2 ) @ ( bNF_We4044943003108391690rdLess @ $o @ A ) ) ) ).
% ctwo_ordLess_Cinfinite
thf(fact_7826_cinfinite__cexp,axiom,
! [A: $tType,B: $tType,Q3: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B )] :
( ( 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_Cardinal_ctwo @ Q3 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 ) )
=> ( bNF_Ca4139267488887388095finite @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R2 ) ) ) ) ).
% cinfinite_cexp
thf(fact_7827_Cinfinite__cexp,axiom,
! [A: $tType,B: $tType,Q3: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B )] :
( ( 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_Cardinal_ctwo @ Q3 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 ) )
=> ( ( bNF_Ca4139267488887388095finite @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R2 ) )
& ( bNF_Ca8970107618336181345der_on @ ( B > A ) @ ( field2 @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R2 ) ) @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R2 ) ) ) ) ) ).
% Cinfinite_cexp
thf(fact_7828_card__of__Plus__Times__aux,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A6: set @ A,B5: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A6 ) )
=> ( ( 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( 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 @ A6 @ B5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times_aux
thf(fact_7829_cprod__cexp__csum__cexp__Cinfinite,axiom,
! [C: $tType,B: $tType,A: $tType,T2: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),S: set @ ( product_prod @ C @ C )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ T2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ T2 ) @ T2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( A > ( sum_sum @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( A > ( sum_sum @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_cexp @ ( product_prod @ B @ C ) @ A @ ( bNF_Cardinal_cprod @ B @ C @ R2 @ S ) @ T2 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ B @ C ) @ A @ ( bNF_Cardinal_csum @ B @ C @ R2 @ S ) @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) ) ).
% cprod_cexp_csum_cexp_Cinfinite
thf(fact_7830_csum__Cfinite__cexp__Cinfinite,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B ),T2: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ( bNF_Cardinal_cfinite @ B @ S )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S ) @ S ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ C @ T2 )
& ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ T2 ) @ T2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ B ) ) ) ) @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ $o ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ B ) ) ) ) @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ $o ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ B ) @ C @ ( bNF_Cardinal_csum @ A @ B @ R2 @ S ) @ T2 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ $o ) @ C @ ( bNF_Cardinal_csum @ A @ $o @ R2 @ bNF_Cardinal_ctwo ) @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) ) ) ) ).
% csum_Cfinite_cexp_Cinfinite
thf(fact_7831_cprod__csum__distrib1,axiom,
! [C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R32: set @ ( product_prod @ C @ C )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_csum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_cprod @ A @ C @ R1 @ R32 ) ) @ ( bNF_Cardinal_cprod @ A @ ( sum_sum @ B @ C ) @ R1 @ ( bNF_Cardinal_csum @ B @ C @ R22 @ R32 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% cprod_csum_distrib1
thf(fact_7832_cprod__csum__cexp,axiom,
! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( $o > ( sum_sum @ A @ B ) ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( $o > ( sum_sum @ A @ B ) ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ B ) @ $o @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ bNF_Cardinal_ctwo ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) ).
% cprod_csum_cexp
thf(fact_7833_card__of__Plus__commute,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B5 @ A6 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_commute
thf(fact_7834_card__of__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A6: set @ A,B5: set @ B,C4: set @ 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ C4 @ A6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C4 @ B5 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% card_of_Plus_mono2
thf(fact_7835_card__of__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A6: set @ A,B5: set @ B,C4: set @ 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A6 @ C4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B5 @ C4 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_mono1
thf(fact_7836_card__of__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,A6: set @ A,B5: set @ B,C4: set @ 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ C4 @ A6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C4 @ B5 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% card_of_Plus_cong2
thf(fact_7837_card__of__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,A6: set @ A,B5: set @ B,C4: set @ 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A6 @ C4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B5 @ C4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_cong1
thf(fact_7838_card__of__Plus__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,A6: set @ A,B5: set @ B,C4: set @ C] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_Plus @ ( sum_sum @ A @ B ) @ C @ ( sum_Plus @ A @ B @ A6 @ B5 ) @ C4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_Plus @ A @ ( sum_sum @ B @ C ) @ A6 @ ( sum_Plus @ B @ C @ B5 @ C4 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_assoc
thf(fact_7839_card__of__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A6: set @ A,B5: set @ B,C4: set @ C,D4: set @ 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ C4 ) @ ( bNF_Ca6860139660246222851ard_of @ D @ D4 ) ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A6 @ C4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B5 @ D4 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% card_of_Plus_mono
thf(fact_7840_card__of__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A6: set @ A,B5: set @ B,C4: set @ C,D4: set @ 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordIso @ 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ C4 ) @ ( bNF_Ca6860139660246222851ard_of @ D @ D4 ) ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A6 @ C4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B5 @ D4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% card_of_Plus_cong
thf(fact_7841_card__of__Plus2,axiom,
! [B: $tType,A: $tType,B5: set @ A,A6: set @ B] : ( 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 @ B5 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A6 @ B5 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus2
thf(fact_7842_card__of__Plus1,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] : ( 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B5 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus1
thf(fact_7843_card__of__Times__Plus__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,A6: set @ A,B5: set @ B,C4: set @ C] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ ( sum_sum @ B @ C ) )
@ ( product_Sigma @ A @ ( sum_sum @ B @ C ) @ A6
@ ^ [Uu3: A] : ( sum_Plus @ B @ C @ B5 @ C4 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) )
@ ( sum_Plus @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 )
@ ( product_Sigma @ A @ C @ A6
@ ^ [Uu3: A] : C4 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Times_Plus_distrib
thf(fact_7844_card__of__Plus__Times__bool,axiom,
! [A: $tType,A6: set @ A] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ $o ) @ ( product_prod @ A @ $o ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ $o ) @ ( product_prod @ A @ $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A6 @ A6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ $o )
@ ( product_Sigma @ A @ $o @ A6
@ ^ [Uu3: A] : ( top_top @ ( set @ $o ) ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ A ) @ ( product_prod @ A @ $o ) ) ) ).
% card_of_Plus_Times_bool
thf(fact_7845_card__of__Un__Plus__ordLeq,axiom,
! [A: $tType,A6: set @ A,B5: 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 ) @ A6 @ B5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A6 @ B5 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% card_of_Un_Plus_ordLeq
thf(fact_7846_card__of__Plus__empty2,axiom,
! [B: $tType,A: $tType,A6: 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( bot_bot @ ( set @ B ) ) @ A6 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_empty2
thf(fact_7847_card__of__Plus__empty1,axiom,
! [B: $tType,A: $tType,A6: 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ ( bot_bot @ ( set @ B ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus_empty1
thf(fact_7848_Un__csum,axiom,
! [A: $tType,A6: set @ A,B5: 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 ) @ A6 @ B5 ) ) @ ( bNF_Cardinal_csum @ A @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B5 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% Un_csum
thf(fact_7849_Plus__csum,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B5 ) ) @ ( bNF_Cardinal_csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ).
% Plus_csum
thf(fact_7850_csum__com,axiom,
! [B: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),P22: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Cardinal_csum @ A @ B @ P1 @ P22 ) @ ( bNF_Cardinal_csum @ B @ A @ P22 @ P1 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ B @ A ) ) ) ).
% csum_com
thf(fact_7851_csum__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: 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 ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordIso @ 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 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_csum @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% csum_cong
thf(fact_7852_csum__csum,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R32: set @ ( product_prod @ C @ C ),R42: set @ ( product_prod @ D @ D )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_csum @ C @ D @ R32 @ R42 ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) @ ( bNF_Cardinal_csum @ A @ C @ R1 @ R32 ) @ ( bNF_Cardinal_csum @ B @ D @ R22 @ R42 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ).
% csum_csum
thf(fact_7853_csum__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: 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 ) ) @ P1 @ R1 ) @ ( 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 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_csum @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% csum_mono
thf(fact_7854_csum__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),P22: set @ ( product_prod @ B @ B ),P32: set @ ( product_prod @ C @ C )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ B ) @ C @ ( bNF_Cardinal_csum @ A @ B @ P1 @ P22 ) @ P32 ) @ ( bNF_Cardinal_csum @ A @ ( sum_sum @ B @ C ) @ P1 @ ( bNF_Cardinal_csum @ B @ C @ P22 @ P32 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% csum_assoc
thf(fact_7855_csum__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: 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 ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P1 @ Q3 ) @ ( bNF_Cardinal_csum @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% csum_cong1
thf(fact_7856_csum__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: 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 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Cardinal_csum @ C @ A @ Q3 @ P22 ) @ ( bNF_Cardinal_csum @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% csum_cong2
thf(fact_7857_csum__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: 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 ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P1 @ Q3 ) @ ( bNF_Cardinal_csum @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% csum_mono1
thf(fact_7858_csum__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: 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 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Cardinal_csum @ C @ A @ Q3 @ P22 ) @ ( bNF_Cardinal_csum @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% csum_mono2
thf(fact_7859_ordLeq__csum2,axiom,
! [B: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),P1: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P22 ) @ P22 )
=> ( 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 ) ) ) @ P22 @ ( bNF_Cardinal_csum @ B @ A @ P1 @ P22 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ) ).
% ordLeq_csum2
thf(fact_7860_ordLeq__csum1,axiom,
! [B: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),P22: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P1 ) @ P1 )
=> ( 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 ) ) ) @ P1 @ ( bNF_Cardinal_csum @ A @ B @ P1 @ P22 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ) ).
% ordLeq_csum1
thf(fact_7861_Card__order__Plus1,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( 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 ) ) ) @ R2 @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ ( field2 @ A @ R2 ) @ B5 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ) ).
% Card_order_Plus1
thf(fact_7862_Card__order__Plus2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( 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 ) ) ) @ R2 @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A6 @ ( field2 @ A @ R2 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ) ).
% Card_order_Plus2
thf(fact_7863_ordLeq__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),A6: set @ 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ A6 @ ( field2 @ A @ R2 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A6 @ ( field2 @ B @ R4 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% ordLeq_Plus_mono2
thf(fact_7864_ordLeq__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),C4: set @ 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R2 ) @ C4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R4 ) @ C4 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% ordLeq_Plus_mono1
thf(fact_7865_ordIso__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),A6: set @ 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ A6 @ ( field2 @ A @ R2 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A6 @ ( field2 @ B @ R4 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% ordIso_Plus_cong2
thf(fact_7866_ordIso__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),C4: set @ 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R2 ) @ C4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R4 ) @ C4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% ordIso_Plus_cong1
thf(fact_7867_ordLeq__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),P2: set @ ( product_prod @ C @ C ),P8: 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 ) ) @ R2 @ R4 ) @ ( 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 ) ) @ P2 @ P8 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R2 ) @ ( field2 @ C @ P2 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R4 ) @ ( field2 @ D @ P8 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% ordLeq_Plus_mono
thf(fact_7868_ordIso__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),P2: set @ ( product_prod @ C @ C ),P8: 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 ) ) @ R2 @ R4 ) @ ( bNF_Wellorder_ordIso @ 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 ) ) @ P2 @ P8 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R2 ) @ ( field2 @ C @ P2 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R4 ) @ ( field2 @ D @ P8 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% ordIso_Plus_cong
thf(fact_7869_card__Plus,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B5 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B5 ) ) ) ) ) ).
% card_Plus
thf(fact_7870_csum__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),P2: set @ ( product_prod @ B @ B ),P8: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca4139267488887388095finite @ A @ R2 )
=> ( ( 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 @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Cardinal_csum @ B @ C @ P2 @ P8 ) @ ( bNF_Cardinal_csum @ A @ A @ R2 @ R2 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ C ) @ ( sum_sum @ A @ 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_Cardinal_csum @ B @ C @ P2 @ P8 ) @ R2 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% csum_dup
thf(fact_7871_card__Plus__conv__if,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] :
( ( ( ( finite_finite2 @ A @ A6 )
& ( finite_finite2 @ B @ B5 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B5 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B5 ) ) ) )
& ( ~ ( ( finite_finite2 @ A @ A6 )
& ( finite_finite2 @ B @ B5 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B5 ) )
= ( zero_zero @ nat ) ) ) ) ).
% card_Plus_conv_if
thf(fact_7872_card__of__Plus__infinite2,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: set @ B] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( 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 @ B5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( 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 @ B5 @ A6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ).
% card_of_Plus_infinite2
thf(fact_7873_card__of__Plus__infinite1,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ B] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( 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 @ B5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( 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 @ A6 @ B5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% card_of_Plus_infinite1
thf(fact_7874_card__of__Plus__infinite,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: set @ B] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( 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 @ B5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( 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 @ A6 @ B5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( 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 @ B5 @ A6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Plus_infinite
thf(fact_7875_card__of__Plus__ordLess__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,C4: set @ A,A6: set @ B,B5: set @ C] :
( ~ ( finite_finite2 @ A @ C4 )
=> ( ( 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C4 ) ) @ ( 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 @ B5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C4 ) ) @ ( 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 @ A6 @ B5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C4 ) ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite
thf(fact_7876_Card__order__Plus__infinite,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),P2: 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 ) ) @ P2 @ 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 @ P2 ) ) ) @ 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 @ P2 ) @ ( field2 @ A @ R2 ) ) ) @ R2 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ) ).
% Card_order_Plus_infinite
thf(fact_7877_csum__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P2: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),Q3: 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 ) ) @ P2 @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q3 @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P2 ) @ P2 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_csum @ A @ C @ P2 @ Q3 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ B ) ) ) ) ) ) ) ).
% csum_cinfinite_bound
thf(fact_7878_csum__absorb2_H,axiom,
! [A: $tType,B: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 )
=> ( ( 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 ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R1 )
| ( bNF_Ca4139267488887388095finite @ A @ R22 ) )
=> ( 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_Cardinal_csum @ B @ A @ R1 @ R22 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% csum_absorb2'
thf(fact_7879_csum__absorb1_H,axiom,
! [B: $tType,A: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 )
=> ( ( 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 ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R1 )
| ( bNF_Ca4139267488887388095finite @ A @ R22 ) )
=> ( 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_Cardinal_csum @ A @ B @ R22 @ R1 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ) ).
% csum_absorb1'
thf(fact_7880_csum__absorb1,axiom,
! [B: $tType,A: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R22 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 ) )
=> ( ( 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 ) ) @ R1 @ R22 ) @ ( 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_Cardinal_csum @ A @ B @ R22 @ R1 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% csum_absorb1
thf(fact_7881_csum__Cnotzero1,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( ~ ( 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 ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
=> ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_czero @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) )
& ( bNF_Ca8970107618336181345der_on @ ( sum_sum @ A @ B ) @ ( field2 @ ( sum_sum @ A @ B ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) ) ) ) ).
% csum_Cnotzero1
thf(fact_7882_card__of__Plus__Times,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A6: set @ A,B1: B,B22: B,B5: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A6 ) )
=> ( ( ( B1 != B22 )
& ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ B1 @ ( insert2 @ B @ B22 @ ( bot_bot @ ( set @ B ) ) ) ) @ B5 ) )
=> ( 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 @ A6 @ B5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B5 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times
thf(fact_7883_card__of__Plus__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ B,B5: 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 @ A6 ) @ 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 @ B5 ) @ 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 @ A6 @ B5 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Plus_ordLeq_infinite_Field
thf(fact_7884_card__of__Plus__ordLess__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ B,B5: 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 @ A6 ) @ 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 @ B5 ) @ 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 @ A6 @ B5 ) ) @ R2 ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite_Field
thf(fact_7885_Plus__eq__empty__conv,axiom,
! [A: $tType,B: $tType,A6: set @ A,B5: set @ B] :
( ( ( sum_Plus @ A @ B @ A6 @ B5 )
= ( bot_bot @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( A6
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Plus_eq_empty_conv
thf(fact_7886_relImage__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Gr4221423524335903396lImage @ B @ A )
= ( ^ [R6: set @ ( product_prod @ B @ B ),F4: B > A] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [A12: B,A23: B] :
( ( Uu3
= ( product_Pair @ A @ A @ ( F4 @ A12 ) @ ( F4 @ A23 ) ) )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A12 @ A23 ) @ R6 ) ) ) ) ) ).
% relImage_def
thf(fact_7887_card__def,axiom,
! [B: $tType] :
( ( finite_card @ B )
= ( finite_folding_F @ B @ nat
@ ^ [Uu3: B] : suc
@ ( zero_zero @ nat ) ) ) ).
% card_def
thf(fact_7888_mono__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ D )
& ( order @ C )
& ( order @ A ) )
=> ! [A6: A > B > $o,B5: C > D > $o] :
( ( bi_total @ A @ B @ A6 )
=> ( ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A6
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A6
@ ^ [Y5: $o,Z: $o] : Y5 = Z )
@ ( ord_less_eq @ A )
@ ( ord_less_eq @ B ) )
=> ( ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B5
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ B5
@ ^ [Y5: $o,Z: $o] : Y5 = Z )
@ ( ord_less_eq @ C )
@ ( ord_less_eq @ D ) )
=> ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A6 @ B5 )
@ ^ [Y5: $o,Z: $o] : Y5 = Z
@ ( order_mono @ A @ C )
@ ( order_mono @ B @ D ) ) ) ) ) ) ).
% mono_transfer
thf(fact_7889_folding__on_Oinsert__remove,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X3: A,A6: set @ A,Z2: B] :
( ( finite_folding_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X3 @ A6 ) @ S3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z2 @ ( insert2 @ A @ X3 @ A6 ) )
= ( F3 @ X3 @ ( finite_folding_F @ A @ B @ F3 @ Z2 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% folding_on.insert_remove
thf(fact_7890_folding__on_Oremove,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,A6: set @ A,X3: A,Z2: B] :
( ( finite_folding_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ S3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z2 @ A6 )
= ( F3 @ X3 @ ( finite_folding_F @ A @ B @ F3 @ Z2 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% folding_on.remove
thf(fact_7891_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_7892_folding__on_Oempty,axiom,
! [A: $tType,B: $tType,S3: set @ A,F3: A > B > B,Z2: B] :
( ( finite_folding_on @ A @ B @ S3 @ F3 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z2 @ ( bot_bot @ ( set @ A ) ) )
= Z2 ) ) ).
% folding_on.empty
thf(fact_7893_folding__on_Oinsert,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X3: A,A6: set @ A,Z2: B] :
( ( finite_folding_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X3 @ A6 ) @ S3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X3 @ A6 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z2 @ ( insert2 @ A @ X3 @ A6 ) )
= ( F3 @ X3 @ ( finite_folding_F @ A @ B @ F3 @ Z2 @ A6 ) ) ) ) ) ) ) ).
% folding_on.insert
thf(fact_7894_folding__idem__on_Oinsert__idem,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X3: A,A6: set @ A,Z2: B] :
( ( finite1890593828518410140dem_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X3 @ A6 ) @ S3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z2 @ ( insert2 @ A @ X3 @ A6 ) )
= ( F3 @ X3 @ ( finite_folding_F @ A @ B @ F3 @ Z2 @ A6 ) ) ) ) ) ) ).
% folding_idem_on.insert_idem
thf(fact_7895_Range__Union,axiom,
! [A: $tType,B: $tType,S3: set @ ( set @ ( product_prod @ B @ A ) )] :
( ( range @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ S3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ B @ A ) ) @ ( set @ A ) @ ( range @ B @ A ) @ S3 ) ) ) ).
% Range_Union
thf(fact_7896_Range__Id__on,axiom,
! [A: $tType,A6: set @ A] :
( ( range @ A @ A @ ( id_on @ A @ A6 ) )
= A6 ) ).
% Range_Id_on
thf(fact_7897_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_7898_Range__Id,axiom,
! [A: $tType] :
( ( range @ A @ A @ ( id2 @ A ) )
= ( top_top @ ( set @ A ) ) ) ).
% Range_Id
thf(fact_7899_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] :
? [X4: B] : ( P @ X4 @ Y3 ) ) ) ).
% Range_Collect_case_prod
thf(fact_7900_Range__insert,axiom,
! [A: $tType,B: $tType,A3: B,B2: A,R2: set @ ( product_prod @ B @ A )] :
( ( range @ B @ A @ ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R2 ) )
= ( insert2 @ A @ B2 @ ( range @ B @ A @ R2 ) ) ) ).
% Range_insert
thf(fact_7901_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_7902_RangeE,axiom,
! [A: $tType,B: $tType,B2: A,R2: set @ ( product_prod @ B @ A )] :
( ( member @ A @ B2 @ ( range @ B @ A @ R2 ) )
=> ~ ! [A5: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A5 @ B2 ) @ R2 ) ) ).
% RangeE
thf(fact_7903_Range_Ointros,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R2 )
=> ( member @ B @ B2 @ ( range @ A @ B @ R2 ) ) ) ).
% Range.intros
thf(fact_7904_Range_Osimps,axiom,
! [B: $tType,A: $tType,A3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A3 @ ( range @ A @ B @ R2 ) )
= ( ? [A8: A,B8: B] :
( ( A3 = B8 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B8 ) @ R2 ) ) ) ) ).
% Range.simps
thf(fact_7905_Range_Ocases,axiom,
! [B: $tType,A: $tType,A3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A3 @ ( range @ A @ B @ R2 ) )
=> ~ ! [A5: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ A3 ) @ R2 ) ) ).
% Range.cases
thf(fact_7906_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_7907_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_7908_Range__Un__eq,axiom,
! [A: $tType,B: $tType,A6: set @ ( product_prod @ B @ A ),B5: set @ ( product_prod @ B @ A )] :
( ( range @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ A6 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( range @ B @ A @ A6 ) @ ( range @ B @ A @ B5 ) ) ) ).
% Range_Un_eq
thf(fact_7909_Range__mono,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S )
=> ( ord_less_eq @ ( set @ B ) @ ( range @ A @ B @ R2 ) @ ( range @ A @ B @ S ) ) ) ).
% Range_mono
thf(fact_7910_Range__snd,axiom,
! [A: $tType,B: $tType] :
( ( range @ B @ A )
= ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) ) ) ).
% Range_snd
thf(fact_7911_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_7912_Range__Int__subset,axiom,
! [A: $tType,B: $tType,A6: set @ ( product_prod @ B @ A ),B5: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( range @ B @ A @ ( inf_inf @ ( set @ ( product_prod @ B @ A ) ) @ A6 @ B5 ) ) @ ( inf_inf @ ( set @ A ) @ ( range @ B @ A @ A6 ) @ ( range @ B @ A @ B5 ) ) ) ).
% Range_Int_subset
thf(fact_7913_Range__Diff__subset,axiom,
! [A: $tType,B: $tType,A6: set @ ( product_prod @ B @ A ),B5: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( range @ B @ A @ A6 ) @ ( range @ B @ A @ B5 ) ) @ ( range @ B @ A @ ( minus_minus @ ( set @ ( product_prod @ B @ A ) ) @ A6 @ B5 ) ) ) ).
% Range_Diff_subset
thf(fact_7914_wf__UN,axiom,
! [B: $tType,A: $tType,I5: set @ A,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( wf @ B @ ( R2 @ I3 ) ) )
=> ( ! [I3: A,J2: A] :
( ( member @ A @ I3 @ I5 )
=> ( ( member @ A @ J2 @ I5 )
=> ( ( ( R2 @ I3 )
!= ( R2 @ J2 ) )
=> ( ( inf_inf @ ( set @ B ) @ ( domain @ B @ B @ ( R2 @ I3 ) ) @ ( 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_7915_wf__Union,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ! [X5: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X5 @ R )
=> ( wf @ A @ X5 ) )
=> ( ! [X5: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X5 @ R )
=> ! [Xa3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa3 @ R )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ X5 ) @ ( range @ A @ A @ Xa3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( wf @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R ) ) ) ) ).
% wf_Union
thf(fact_7916_Domain__Id__on,axiom,
! [A: $tType,A6: set @ A] :
( ( domain @ A @ A @ ( id_on @ A @ A6 ) )
= A6 ) ).
% Domain_Id_on
thf(fact_7917_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_7918_Domain__Id,axiom,
! [A: $tType] :
( ( domain @ A @ A @ ( id2 @ A ) )
= ( top_top @ ( set @ A ) ) ) ).
% Domain_Id
thf(fact_7919_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
@ ^ [X4: A] :
? [X8: B] : ( P @ X4 @ X8 ) ) ) ).
% Domain_Collect_case_prod
thf(fact_7920_Domain__insert,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R2: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R2 ) )
= ( insert2 @ A @ A3 @ ( domain @ A @ B @ R2 ) ) ) ).
% Domain_insert
thf(fact_7921_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_7922_Domain__fst,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) ).
% Domain_fst
thf(fact_7923_Domain__mono,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S )
=> ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R2 ) @ ( domain @ A @ B @ S ) ) ) ).
% Domain_mono
thf(fact_7924_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_7925_Domain__Un__eq,axiom,
! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B ),B5: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( domain @ A @ B @ A6 ) @ ( domain @ A @ B @ B5 ) ) ) ).
% Domain_Un_eq
thf(fact_7926_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_7927_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
=> ~ ! [B4: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B4 ) @ R2 ) ) ).
% Domain.cases
thf(fact_7928_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
= ( ? [A8: A,B8: B] :
( ( A3 = A8 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B8 ) @ R2 ) ) ) ) ).
% Domain.simps
thf(fact_7929_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R2 )
=> ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) ) ) ).
% Domain.DomainI
thf(fact_7930_DomainE,axiom,
! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
=> ~ ! [B4: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B4 ) @ R2 ) ) ).
% DomainE
thf(fact_7931_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_7932_Not__Domain__rtrancl,axiom,
! [A: $tType,X3: A,R: set @ ( product_prod @ A @ A ),Y: A] :
( ~ ( member @ A @ X3 @ ( domain @ A @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R ) )
= ( X3 = Y ) ) ) ).
% Not_Domain_rtrancl
thf(fact_7933_Domain__unfold,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ^ [X4: A] :
? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R5 ) ) ) ) ).
% Domain_unfold
thf(fact_7934_Domain__Int__subset,axiom,
! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B ),B5: set @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ B5 ) ) @ ( inf_inf @ ( set @ A ) @ ( domain @ A @ B @ A6 ) @ ( domain @ A @ B @ B5 ) ) ) ).
% Domain_Int_subset
thf(fact_7935_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_7936_Domain__Diff__subset,axiom,
! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B ),B5: set @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ B @ A6 ) @ ( domain @ A @ B @ B5 ) ) @ ( domain @ A @ B @ ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ B5 ) ) ) ).
% Domain_Diff_subset
thf(fact_7937_Domain__Union,axiom,
! [B: $tType,A: $tType,S3: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( domain @ A @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ A ) @ ( domain @ A @ B ) @ S3 ) ) ) ).
% Domain_Union
thf(fact_7938_wf__Un,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( ( wf @ A @ S )
=> ( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R2 ) @ ( range @ A @ A @ S ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S ) ) ) ) ) ).
% wf_Un
thf(fact_7939_irrefl__tranclI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A] :
( ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R2 ) @ ( transitive_rtrancl @ A @ R2 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ).
% irrefl_tranclI
thf(fact_7940_prod__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,X3: A,Y: B] :
( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y ) )
= ( insert2 @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) ) ).
% prod_set_simps(2)
thf(fact_7941_converse__inject,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A )] :
( ( ( converse @ B @ A @ R2 )
= ( converse @ B @ A @ S ) )
= ( R2 = S ) ) ).
% converse_inject
thf(fact_7942_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_7943_converse__iff,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R2: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( converse @ B @ A @ R2 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ R2 ) ) ).
% converse_iff
thf(fact_7944_Field__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( converse @ A @ A @ R2 ) )
= ( field2 @ A @ R2 ) ) ).
% Field_converse
thf(fact_7945_converse__mono,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S ) ) ).
% converse_mono
thf(fact_7946_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_7947_converse__Id,axiom,
! [A: $tType] :
( ( converse @ A @ A @ ( id2 @ A ) )
= ( id2 @ A ) ) ).
% converse_Id
thf(fact_7948_refl__on__converse,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A6 @ ( converse @ A @ A @ R2 ) )
= ( refl_on @ A @ A6 @ R2 ) ) ).
% refl_on_converse
thf(fact_7949_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_7950_total__on__converse,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ A6 @ ( converse @ A @ A @ R2 ) )
= ( total_on @ A @ A6 @ R2 ) ) ).
% total_on_converse
thf(fact_7951_antisym__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( antisym @ A @ ( converse @ A @ A @ R2 ) )
= ( antisym @ A @ R2 ) ) ).
% antisym_converse
thf(fact_7952_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_7953_converse__Id__on,axiom,
! [A: $tType,A6: set @ A] :
( ( converse @ A @ A @ ( id_on @ A @ A6 ) )
= ( id_on @ A @ A6 ) ) ).
% converse_Id_on
thf(fact_7954_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_7955_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_7956_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_7957_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_7958_rtrancl__converseI,axiom,
! [A: $tType,Y: A,X3: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R2 ) ) ) ) ).
% rtrancl_converseI
thf(fact_7959_rtrancl__converseD,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R2 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ).
% rtrancl_converseD
thf(fact_7960_trancl__converseI,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R2 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R2 ) ) ) ) ).
% trancl_converseI
thf(fact_7961_trancl__converseD,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R2 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_converseD
thf(fact_7962_converseI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R2 )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ ( converse @ A @ B @ R2 ) ) ) ).
% converseI
thf(fact_7963_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 ) )
=> ~ ! [X5: B,Y4: A] :
( ( Yx
= ( product_Pair @ A @ B @ Y4 @ X5 ) )
=> ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X5 @ Y4 ) @ R2 ) ) ) ).
% converseE
thf(fact_7964_converseD,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R2: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( converse @ B @ A @ R2 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ R2 ) ) ).
% converseD
thf(fact_7965_converse_Osimps,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R2 ) )
= ( ? [A8: A,B8: B] :
( ( A1 = B8 )
& ( A22 = A8 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B8 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_7966_converse_Ocases,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R2 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A22 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_7967_in__listrel1__converse,axiom,
! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( listrel1 @ A @ ( converse @ A @ A @ R2 ) ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( converse @ ( list @ A ) @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ).
% in_listrel1_converse
thf(fact_7968_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,X4: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y3 ) @ R5 ) ) ) ) ) ).
% converse_unfold
thf(fact_7969_conversep__converse__eq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( conversep @ A @ B
@ ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R2 ) )
= ( ^ [X4: B,Y3: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y3 ) @ ( converse @ A @ B @ R2 ) ) ) ) ).
% conversep_converse_eq
thf(fact_7970_converse__def,axiom,
! [B: $tType,A: $tType] :
( ( converse @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ( conversep @ A @ B
@ ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R5 ) ) ) ) ) ) ).
% converse_def
thf(fact_7971_converse__Int,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A )] :
( ( converse @ B @ A @ ( inf_inf @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S ) ) ) ).
% converse_Int
thf(fact_7972_converse__relcomp,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ B @ C ),S: set @ ( product_prod @ C @ A )] :
( ( converse @ B @ A @ ( relcomp @ B @ C @ A @ R2 @ S ) )
= ( relcomp @ A @ C @ B @ ( converse @ C @ A @ S ) @ ( converse @ B @ C @ R2 ) ) ) ).
% converse_relcomp
thf(fact_7973_converse__Un,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A )] :
( ( converse @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S ) ) ) ).
% converse_Un
thf(fact_7974_Image__subset__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A ),A6: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R2 @ A6 ) @ B5 )
= ( ord_less_eq @ ( set @ B ) @ A6 @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ ( converse @ B @ A @ R2 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) ) ) ) ) ).
% Image_subset_eq
thf(fact_7975_converse__subset__swap,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ ( converse @ B @ A @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( converse @ A @ B @ R2 ) @ S ) ) ).
% converse_subset_swap
thf(fact_7976_converse__UNION,axiom,
! [B: $tType,A: $tType,C: $tType,R2: C > ( set @ ( product_prod @ B @ A ) ),S3: set @ C] :
( ( converse @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S3 ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: C] : ( converse @ B @ A @ ( R2 @ X4 ) )
@ S3 ) ) ) ).
% converse_UNION
thf(fact_7977_converse__INTER,axiom,
! [B: $tType,A: $tType,C: $tType,R2: C > ( set @ ( product_prod @ B @ A ) ),S3: set @ C] :
( ( converse @ B @ A @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S3 ) ) )
= ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: C] : ( converse @ B @ A @ ( R2 @ X4 ) )
@ S3 ) ) ) ).
% converse_INTER
thf(fact_7978_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_7979_Image__INT__eq,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ B @ A ),A6: set @ C,B5: C > ( set @ B )] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R2 ) )
=> ( ( A6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( image @ B @ A @ R2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B5 @ A6 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( image @ B @ A @ R2 @ ( B5 @ X4 ) )
@ A6 ) ) ) ) ) ).
% Image_INT_eq
thf(fact_7980_single__valued__subset,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S )
=> ( ( single_valued @ A @ B @ S )
=> ( single_valued @ A @ B @ R2 ) ) ) ).
% single_valued_subset
thf(fact_7981_single__valued__Id__on,axiom,
! [A: $tType,A6: set @ A] : ( single_valued @ A @ A @ ( id_on @ A @ A6 ) ) ).
% single_valued_Id_on
thf(fact_7982_single__valued__relcomp,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( single_valued @ A @ B @ R2 )
=> ( ( single_valued @ B @ C @ S )
=> ( single_valued @ A @ C @ ( relcomp @ A @ B @ C @ R2 @ S ) ) ) ) ).
% single_valued_relcomp
thf(fact_7983_single__valued__Id,axiom,
! [A: $tType] : ( single_valued @ A @ A @ ( id2 @ A ) ) ).
% single_valued_Id
thf(fact_7984_single__valued__empty,axiom,
! [B: $tType,A: $tType] : ( single_valued @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% single_valued_empty
thf(fact_7985_single__valued__def,axiom,
! [B: $tType,A: $tType] :
( ( single_valued @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
! [X4: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R5 )
=> ! [Z4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Z4 ) @ R5 )
=> ( Y3 = Z4 ) ) ) ) ) ).
% single_valued_def
thf(fact_7986_single__valuedI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ! [X5: A,Y4: B,Z3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Z3 ) @ R2 )
=> ( Y4 = Z3 ) ) )
=> ( single_valued @ A @ B @ R2 ) ) ).
% single_valuedI
thf(fact_7987_single__valuedD,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),X3: A,Y: B,Z2: B] :
( ( single_valued @ A @ B @ R2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z2 ) @ R2 )
=> ( Y = Z2 ) ) ) ) ).
% single_valuedD
thf(fact_7988_single__valued__confluent,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A,Z2: A] :
( ( single_valued @ A @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z2 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ).
% single_valued_confluent
thf(fact_7989_Image__Int__eq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),A6: set @ B,B5: set @ B] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R ) )
=> ( ( image @ B @ A @ R @ ( inf_inf @ ( set @ B ) @ A6 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ R @ A6 ) @ ( image @ B @ A @ R @ B5 ) ) ) ) ).
% Image_Int_eq
thf(fact_7990_trans__wf__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( ( wf @ A @ R2 )
= ( ! [A8: A] :
( wf @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( image @ A @ A @ ( converse @ A @ A @ R2 ) @ ( insert2 @ A @ A8 @ ( bot_bot @ ( set @ A ) ) ) )
@ ^ [Uu3: A] : ( image @ A @ A @ ( converse @ A @ A @ R2 ) @ ( insert2 @ A @ A8 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% trans_wf_iff
thf(fact_7991_single__valuedp__single__valued__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( single_valuedp @ A @ B
@ ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R2 ) )
= ( single_valued @ A @ B @ R2 ) ) ).
% single_valuedp_single_valued_eq
thf(fact_7992_trans__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ ( converse @ A @ A @ R2 ) )
= ( trans @ A @ R2 ) ) ).
% trans_converse
thf(fact_7993_transp__trans__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( transp @ A
@ ^ [X4: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 ) )
= ( trans @ A @ R2 ) ) ).
% transp_trans_eq
thf(fact_7994_lexord__trans,axiom,
! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A ),Z2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z2 ) @ ( lexord @ A @ R2 ) )
=> ( ( trans @ A @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Z2 ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_trans
thf(fact_7995_trans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: A,Y3: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z4 ) @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z4 ) @ R5 ) ) ) ) ) ).
% trans_def
thf(fact_7996_transI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X5: A,Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Z3 ) @ R2 ) ) )
=> ( trans @ A @ R2 ) ) ).
% transI
thf(fact_7997_transE,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A,Z2: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ R2 ) ) ) ) ).
% transE
thf(fact_7998_transD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A,Z2: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ R2 ) ) ) ) ).
% transD
thf(fact_7999_lenlex__trans,axiom,
! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A ),Z2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lenlex @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z2 ) @ ( lenlex @ A @ R2 ) )
=> ( ( trans @ A @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Z2 ) @ ( lenlex @ A @ R2 ) ) ) ) ) ).
% lenlex_trans
thf(fact_8000_trans__empty,axiom,
! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trans_empty
thf(fact_8001_single__valuedp__bot,axiom,
! [B: $tType,A: $tType] : ( single_valuedp @ A @ B @ ( bot_bot @ ( A > B > $o ) ) ) ).
% single_valuedp_bot
thf(fact_8002_under__incr,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R2 @ A3 ) @ ( order_under @ A @ R2 @ B2 ) ) ) ) ).
% under_incr
thf(fact_8003_trans__Int,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( ( trans @ A @ S )
=> ( trans @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S ) ) ) ) ).
% trans_Int
thf(fact_8004_trans__Id,axiom,
! [A: $tType] : ( trans @ A @ ( id2 @ A ) ) ).
% trans_Id
thf(fact_8005_trans__Id__on,axiom,
! [A: $tType,A6: set @ A] : ( trans @ A @ ( id_on @ A @ A6 ) ) ).
% trans_Id_on
thf(fact_8006_single__valuedpD,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X3: A,Y: B,Z2: B] :
( ( single_valuedp @ A @ B @ R2 )
=> ( ( R2 @ X3 @ Y )
=> ( ( R2 @ X3 @ Z2 )
=> ( Y = Z2 ) ) ) ) ).
% single_valuedpD
thf(fact_8007_single__valuedpI,axiom,
! [B: $tType,A: $tType,R2: A > B > $o] :
( ! [X5: A,Y4: B,Z3: B] :
( ( R2 @ X5 @ Y4 )
=> ( ( R2 @ X5 @ Z3 )
=> ( Y4 = Z3 ) ) )
=> ( single_valuedp @ A @ B @ R2 ) ) ).
% single_valuedpI
thf(fact_8008_single__valuedp__def,axiom,
! [B: $tType,A: $tType] :
( ( single_valuedp @ A @ B )
= ( ^ [R5: A > B > $o] :
! [X4: A,Y3: B] :
( ( R5 @ X4 @ Y3 )
=> ! [Z4: B] :
( ( R5 @ X4 @ Z4 )
=> ( Y3 = Z4 ) ) ) ) ) ).
% single_valuedp_def
thf(fact_8009_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_8010_single__valuedp__less__eq,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,S: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R2 @ S )
=> ( ( single_valuedp @ A @ B @ S )
=> ( single_valuedp @ A @ B @ R2 ) ) ) ).
% single_valuedp_less_eq
thf(fact_8011_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_8012_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_8013_trans__INTER,axiom,
! [B: $tType,A: $tType,S3: set @ A,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( trans @ B @ ( R2 @ X5 ) ) )
=> ( trans @ B @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S3 ) ) ) ) ).
% trans_INTER
thf(fact_8014_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_8015_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_8016_trans__join,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Y17: A] :
! [Z4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ Z4 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y24: A,Aa3: A] :
( ( Y17 = Y24 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Aa3 ) @ R5 ) )
@ Z4 ) )
@ X4 ) ) ) ) ).
% trans_join
thf(fact_8017_underS__incr,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( trans @ A @ R2 )
=> ( ( antisym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A3 ) @ ( order_underS @ A @ R2 @ B2 ) ) ) ) ) ).
% underS_incr
thf(fact_8018_wf__finite__segments,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R2 )
=> ( ( trans @ A @ R2 )
=> ( ! [X5: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X5 ) @ R2 ) ) )
=> ( wf @ A @ R2 ) ) ) ) ).
% wf_finite_segments
thf(fact_8019_prod__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X3: A,Y: B] :
( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y ) )
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% prod_set_simps(1)
thf(fact_8020_numeral__le__enat__iff,axiom,
! [M2: num,N: nat] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M2 ) @ N ) ) ).
% numeral_le_enat_iff
thf(fact_8021_plus__enat__simps_I1_J,axiom,
! [M2: nat,N: nat] :
( ( plus_plus @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% plus_enat_simps(1)
thf(fact_8022_enat__ord__simps_I1_J,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% enat_ord_simps(1)
thf(fact_8023_numeral__less__enat__iff,axiom,
! [M2: num,N: nat] :
( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( extended_enat2 @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ M2 ) @ N ) ) ).
% numeral_less_enat_iff
thf(fact_8024_iadd__le__enat__iff,axiom,
! [X3: extended_enat,Y: extended_enat,N: nat] :
( ( ord_less_eq @ extended_enat @ ( plus_plus @ extended_enat @ X3 @ Y ) @ ( extended_enat2 @ N ) )
= ( ? [Y7: nat,X9: nat] :
( ( X3
= ( extended_enat2 @ X9 ) )
& ( Y
= ( extended_enat2 @ Y7 ) )
& ( ord_less_eq @ nat @ ( plus_plus @ nat @ X9 @ Y7 ) @ N ) ) ) ) ).
% iadd_le_enat_iff
thf(fact_8025_numeral__eq__enat,axiom,
( ( numeral_numeral @ extended_enat )
= ( ^ [K3: num] : ( extended_enat2 @ ( numeral_numeral @ nat @ K3 ) ) ) ) ).
% numeral_eq_enat
thf(fact_8026_Suc__ile__eq,axiom,
! [M2: nat,N: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ ( suc @ M2 ) ) @ N )
= ( ord_less @ extended_enat @ ( extended_enat2 @ M2 ) @ N ) ) ).
% Suc_ile_eq
thf(fact_8027_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_8028_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_8029_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,L: nat] :
( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ ( extended_enat2 @ L ) )
= ( vEBT_Node @ Info @ Deg
@ ( take @ vEBT_VEBT @ ( divide_divide @ nat @ L @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList ) )
@ ( vEBT_VEBT_elim_dead @ Summary @ ( extended_enat2 @ ( divide_divide @ nat @ L @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.simps(3)
thf(fact_8030_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
! [A3: $o,B2: $o,Uu2: extended_enat] :
( ( vEBT_VEBT_elim_dead @ ( vEBT_Leaf @ A3 @ B2 ) @ Uu2 )
= ( vEBT_Leaf @ A3 @ B2 ) ) ).
% VEBT_internal.elim_dead.simps(1)
thf(fact_8031_VEBT__internal_Oelim__dead_Oelims,axiom,
! [X3: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
( ( ( vEBT_VEBT_elim_dead @ X3 @ Xa2 )
= Y )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y
!= ( vEBT_Leaf @ A5 @ B4 ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( Xa2
= ( 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] :
( ( X3
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) )
=> ! [L4: nat] :
( ( Xa2
= ( extended_enat2 @ L4 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ Deg2
@ ( take @ vEBT_VEBT @ ( divide_divide @ nat @ L4 @ ( 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 @ L4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.elims
thf(fact_8032_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( vEBT_Node @ Info @ Deg
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList )
@ ( vEBT_VEBT_elim_dead @ Summary @ ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% VEBT_internal.elim_dead.simps(2)
thf(fact_8033_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_8034_numeral__ne__infinity,axiom,
! [K2: num] :
( ( numeral_numeral @ extended_enat @ K2 )
!= ( extend4730790105801354508finity @ extended_enat ) ) ).
% numeral_ne_infinity
thf(fact_8035_VEBT__internal_Oelim__dead_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ extended_enat] :
( ! [A5: $o,B4: $o,Uu: extended_enat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Leaf @ A5 @ B4 ) @ Uu ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( X3
!= ( 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,L4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ ( extended_enat2 @ L4 ) ) ) ) ) ).
% VEBT_internal.elim_dead.cases
thf(fact_8036_VEBT__internal_Oelim__dead_Opelims,axiom,
! [X3: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
( ( ( vEBT_VEBT_elim_dead @ X3 @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ X3 @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X3
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) )
=> ( ( Xa2
= ( 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] :
( ( X3
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) )
=> ! [L4: nat] :
( ( Xa2
= ( extended_enat2 @ L4 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ Deg2
@ ( take @ vEBT_VEBT @ ( divide_divide @ nat @ L4 @ ( 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 @ L4 @ ( 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 @ L4 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.pelims
thf(fact_8037_plus__enat__def,axiom,
( ( plus_plus @ extended_enat )
= ( ^ [M5: extended_enat,N3: extended_enat] :
( extended_case_enat @ extended_enat
@ ^ [O: nat] :
( extended_case_enat @ extended_enat
@ ^ [P5: nat] : ( extended_enat2 @ ( plus_plus @ nat @ O @ P5 ) )
@ ( extend4730790105801354508finity @ extended_enat )
@ N3 )
@ ( extend4730790105801354508finity @ extended_enat )
@ M5 ) ) ) ).
% plus_enat_def
thf(fact_8038_eSuc__def,axiom,
( extended_eSuc
= ( extended_case_enat @ extended_enat
@ ^ [N3: nat] : ( extended_enat2 @ ( suc @ N3 ) )
@ ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% eSuc_def
thf(fact_8039_Nats__altdef2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [N3: A] :
( ( member @ A @ N3 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ N3 ) ) ) ) ) ).
% Nats_altdef2
thf(fact_8040_eSuc__numeral,axiom,
! [K2: num] :
( ( extended_eSuc @ ( numeral_numeral @ extended_enat @ K2 ) )
= ( numeral_numeral @ extended_enat @ ( plus_plus @ num @ K2 @ one2 ) ) ) ).
% eSuc_numeral
thf(fact_8041_Nats__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_add
thf(fact_8042_Nats__diff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( member @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ) ).
% Nats_diff
thf(fact_8043_Nats__cases,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X3: A] :
( ( member @ A @ X3 @ ( semiring_1_Nats @ A ) )
=> ~ ! [N2: nat] :
( X3
!= ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% Nats_cases
thf(fact_8044_Nats__induct,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X3: A,P: A > $o] :
( ( member @ A @ X3 @ ( semiring_1_Nats @ A ) )
=> ( ! [N2: nat] : ( P @ ( semiring_1_of_nat @ A @ N2 ) )
=> ( P @ X3 ) ) ) ) ).
% Nats_induct
thf(fact_8045_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_8046_Nats__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_mult
thf(fact_8047_Nats__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [W: num] : ( member @ A @ ( numeral_numeral @ A @ W ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_numeral
thf(fact_8048_Nats__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_1
thf(fact_8049_Nats__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_0
thf(fact_8050_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_8051_enat__eSuc__iff,axiom,
! [Y: nat,X3: extended_enat] :
( ( ( extended_enat2 @ Y )
= ( extended_eSuc @ X3 ) )
= ( ? [N3: nat] :
( ( Y
= ( suc @ N3 ) )
& ( ( extended_enat2 @ N3 )
= X3 ) ) ) ) ).
% enat_eSuc_iff
thf(fact_8052_eSuc__enat__iff,axiom,
! [X3: extended_enat,Y: nat] :
( ( ( extended_eSuc @ X3 )
= ( extended_enat2 @ Y ) )
= ( ? [N3: nat] :
( ( Y
= ( suc @ N3 ) )
& ( X3
= ( extended_enat2 @ N3 ) ) ) ) ) ).
% eSuc_enat_iff
thf(fact_8053_eSuc__enat,axiom,
! [N: nat] :
( ( extended_eSuc @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( suc @ N ) ) ) ).
% eSuc_enat
thf(fact_8054_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_8055_rel__filter_Ocases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,F6: filter @ A,G7: filter @ B] :
( ( rel_filter @ A @ B @ R @ F6 @ G7 )
=> ~ ! [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 )
= F6 )
=> ( ( 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 )
!= G7 ) ) ) ) ).
% rel_filter.cases
thf(fact_8056_rel__filter_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( rel_filter @ A @ B )
= ( ^ [R6: A > B > $o,F9: filter @ A,G9: filter @ B] :
? [Z8: filter @ ( product_prod @ A @ B )] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) @ Z8 )
& ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) @ ( product_fst @ A @ B ) @ Z8 )
= F9 )
& ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) @ ( product_snd @ A @ B ) @ Z8 )
= G9 ) ) ) ) ).
% rel_filter.simps
thf(fact_8057_bot__filter__parametric,axiom,
! [A: $tType,B: $tType,A6: A > B > $o] : ( rel_filter @ A @ B @ A6 @ ( bot_bot @ ( filter @ A ) ) @ ( bot_bot @ ( filter @ B ) ) ) ).
% bot_filter_parametric
thf(fact_8058_rel__filter__eq,axiom,
! [A: $tType] :
( ( rel_filter @ A @ A
@ ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [Y5: filter @ A,Z: filter @ A] : Y5 = Z ) ) ).
% rel_filter_eq
thf(fact_8059_rel__filter__mono,axiom,
! [B: $tType,A: $tType,A6: A > B > $o,B5: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A6 @ B5 )
=> ( ord_less_eq @ ( ( filter @ A ) > ( filter @ B ) > $o ) @ ( rel_filter @ A @ B @ A6 ) @ ( rel_filter @ A @ B @ B5 ) ) ) ).
% rel_filter_mono
thf(fact_8060_rel__filter__conversep,axiom,
! [A: $tType,B: $tType,A6: B > A > $o] :
( ( rel_filter @ A @ B @ ( conversep @ B @ A @ A6 ) )
= ( conversep @ ( filter @ B ) @ ( filter @ A ) @ ( rel_filter @ B @ A @ A6 ) ) ) ).
% rel_filter_conversep
thf(fact_8061_eventually__parametric,axiom,
! [A: $tType,B: $tType,A6: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( filter @ A ) > $o ) @ ( ( filter @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A6
@ ^ [Y5: $o,Z: $o] : Y5 = Z )
@ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ $o @ $o @ ( rel_filter @ A @ B @ A6 )
@ ^ [Y5: $o,Z: $o] : Y5 = Z )
@ ( eventually @ A )
@ ( eventually @ B ) ) ).
% eventually_parametric
thf(fact_8062_filtermap__parametric,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A6: A > C > $o,B5: 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 @ A6 @ B5 ) @ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ C ) @ ( filter @ B ) @ ( filter @ D ) @ ( rel_filter @ A @ C @ A6 ) @ ( rel_filter @ B @ D @ B5 ) ) @ ( filtermap @ A @ B ) @ ( filtermap @ C @ D ) ) ).
% filtermap_parametric
thf(fact_8063_sup__filter__parametric,axiom,
! [A: $tType,B: $tType,A6: A > B > $o] : ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > ( filter @ A ) ) @ ( ( filter @ B ) > ( filter @ B ) ) @ ( rel_filter @ A @ B @ A6 ) @ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A6 ) @ ( rel_filter @ A @ B @ A6 ) ) @ ( sup_sup @ ( filter @ A ) ) @ ( sup_sup @ ( filter @ B ) ) ) ).
% sup_filter_parametric
thf(fact_8064_top__filter__parametric,axiom,
! [A: $tType,B: $tType,A6: A > B > $o] :
( ( bi_total @ A @ B @ A6 )
=> ( rel_filter @ A @ B @ A6 @ ( top_top @ ( filter @ A ) ) @ ( top_top @ ( filter @ B ) ) ) ) ).
% top_filter_parametric
thf(fact_8065_bi__total__rel__filter,axiom,
! [B: $tType,A: $tType,A6: A > B > $o] :
( ( bi_total @ A @ B @ A6 )
=> ( bi_total @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A6 ) ) ) ).
% bi_total_rel_filter
thf(fact_8066_rel__filter_Ointros,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Z7: filter @ ( product_prod @ A @ B ),F6: filter @ A,G7: 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 )
= F6 )
=> ( ( ( 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 )
= G7 )
=> ( rel_filter @ A @ B @ R @ F6 @ G7 ) ) ) ) ).
% rel_filter.intros
thf(fact_8067_Real_Opositive_Orsp,axiom,
( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ $o @ $o @ realrel
@ ^ [Y5: $o,Z: $o] : Y5 = Z
@ ^ [X8: nat > rat] :
? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ R5 @ ( X8 @ N3 ) ) ) )
@ ^ [X8: nat > rat] :
? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ R5 @ ( X8 @ N3 ) ) ) ) ) ).
% Real.positive.rsp
thf(fact_8068_cauchy__def,axiom,
( cauchy
= ( ^ [X8: nat > rat] :
! [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
=> ? [K3: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ K3 @ M5 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) ) @ R5 ) ) ) ) ) ) ).
% cauchy_def
thf(fact_8069_cauchyD,axiom,
! [X6: nat > rat,R2: rat] :
( ( cauchy @ X6 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ? [K: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ K @ M3 )
=> ! [N9: nat] :
( ( ord_less_eq @ nat @ K @ N9 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X6 @ M3 ) @ ( X6 @ N9 ) ) ) @ R2 ) ) ) ) ) ).
% cauchyD
thf(fact_8070_cauchyI,axiom,
! [X6: nat > rat] :
( ! [R3: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R3 )
=> ? [K4: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ K4 @ M )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ K4 @ N2 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X6 @ M ) @ ( X6 @ N2 ) ) ) @ R3 ) ) ) )
=> ( cauchy @ X6 ) ) ).
% cauchyI
thf(fact_8071_le__Real,axiom,
! [X6: nat > rat,Y8: nat > rat] :
( ( cauchy @ X6 )
=> ( ( cauchy @ Y8 )
=> ( ( ord_less_eq @ real @ ( real2 @ X6 ) @ ( real2 @ Y8 ) )
= ( ! [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
=> ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less_eq @ rat @ ( X6 @ N3 ) @ ( plus_plus @ rat @ ( Y8 @ N3 ) @ R5 ) ) ) ) ) ) ) ) ).
% le_Real
thf(fact_8072_cauchy__not__vanishes,axiom,
! [X6: nat > rat] :
( ( cauchy @ X6 )
=> ( ~ ( vanishes @ X6 )
=> ? [B4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ B4 )
& ? [K: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ K @ N9 )
=> ( ord_less @ rat @ B4 @ ( abs_abs @ rat @ ( X6 @ N9 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes
thf(fact_8073_vanishesD,axiom,
! [X6: nat > rat,R2: rat] :
( ( vanishes @ X6 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ? [K: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ K @ N9 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X6 @ N9 ) ) @ R2 ) ) ) ) ).
% vanishesD
thf(fact_8074_vanishesI,axiom,
! [X6: nat > rat] :
( ! [R3: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R3 )
=> ? [K4: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K4 @ N2 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X6 @ N2 ) ) @ R3 ) ) )
=> ( vanishes @ X6 ) ) ).
% vanishesI
thf(fact_8075_vanishes__def,axiom,
( vanishes
= ( ^ [X8: nat > rat] :
! [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
=> ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X8 @ N3 ) ) @ R5 ) ) ) ) ) ).
% vanishes_def
thf(fact_8076_cauchy__not__vanishes__cases,axiom,
! [X6: nat > rat] :
( ( cauchy @ X6 )
=> ( ~ ( vanishes @ X6 )
=> ? [B4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ B4 )
& ? [K: nat] :
( ! [N9: nat] :
( ( ord_less_eq @ nat @ K @ N9 )
=> ( ord_less @ rat @ B4 @ ( uminus_uminus @ rat @ ( X6 @ N9 ) ) ) )
| ! [N9: nat] :
( ( ord_less_eq @ nat @ K @ N9 )
=> ( ord_less @ rat @ B4 @ ( X6 @ N9 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes_cases
thf(fact_8077_not__positive__Real,axiom,
! [X6: nat > rat] :
( ( cauchy @ X6 )
=> ( ( ~ ( positive @ ( real2 @ X6 ) ) )
= ( ! [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
=> ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less_eq @ rat @ ( X6 @ N3 ) @ R5 ) ) ) ) ) ) ).
% not_positive_Real
thf(fact_8078_positive__Real,axiom,
! [X6: nat > rat] :
( ( cauchy @ X6 )
=> ( ( positive @ ( real2 @ X6 ) )
= ( ? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ R5 @ ( X6 @ N3 ) ) ) ) ) ) ) ).
% positive_Real
thf(fact_8079_Real_Opositive_Oabs__eq,axiom,
! [X3: nat > rat] :
( ( realrel @ X3 @ X3 )
=> ( ( positive @ ( real2 @ X3 ) )
= ( ? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ R5 @ ( X3 @ N3 ) ) ) ) ) ) ) ).
% Real.positive.abs_eq
thf(fact_8080_Real_Opositive_Otransfer,axiom,
( bNF_rel_fun @ ( nat > rat ) @ real @ $o @ $o @ pcr_real
@ ^ [Y5: $o,Z: $o] : Y5 = Z
@ ^ [X8: nat > rat] :
? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ R5 @ ( X8 @ N3 ) ) ) )
@ positive ) ).
% Real.positive.transfer
thf(fact_8081_Real_Opositive_Orep__eq,axiom,
( positive
= ( ^ [X4: real] :
? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ R5 @ ( rep_real @ X4 @ N3 ) ) ) ) ) ) ).
% Real.positive.rep_eq
thf(fact_8082_inf__filter__parametric,axiom,
! [A: $tType,B: $tType,A6: A > B > $o] :
( ( bi_unique @ A @ B @ A6 )
=> ( ( bi_total @ A @ B @ A6 )
=> ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > ( filter @ A ) ) @ ( ( filter @ B ) > ( filter @ B ) ) @ ( rel_filter @ A @ B @ A6 ) @ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A6 ) @ ( rel_filter @ A @ B @ A6 ) ) @ ( inf_inf @ ( filter @ A ) ) @ ( inf_inf @ ( filter @ B ) ) ) ) ) ).
% inf_filter_parametric
thf(fact_8083_frequently__parametric,axiom,
! [A: $tType,B: $tType,A6: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( filter @ A ) > $o ) @ ( ( filter @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A6
@ ^ [Y5: $o,Z: $o] : Y5 = Z )
@ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ $o @ $o @ ( rel_filter @ A @ B @ A6 )
@ ^ [Y5: $o,Z: $o] : Y5 = Z )
@ ( frequently @ A )
@ ( frequently @ B ) ) ).
% frequently_parametric
thf(fact_8084_frequently__const,axiom,
! [A: $tType,F6: filter @ A,P: $o] :
( ( F6
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( frequently @ A
@ ^ [X4: A] : P
@ F6 )
= P ) ) ).
% frequently_const
thf(fact_8085_bi__unique__rel__filter,axiom,
! [B: $tType,A: $tType,A6: A > B > $o] :
( ( bi_unique @ A @ B @ A6 )
=> ( bi_unique @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A6 ) ) ) ).
% bi_unique_rel_filter
thf(fact_8086_frequently__bex__finite,axiom,
! [A: $tType,B: $tType,A6: set @ A,P: B > A > $o,F6: filter @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( frequently @ B
@ ^ [X4: B] :
? [Y3: A] :
( ( member @ A @ Y3 @ A6 )
& ( P @ X4 @ Y3 ) )
@ F6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A6 )
& ( frequently @ B
@ ^ [Y3: B] : ( P @ Y3 @ X5 )
@ F6 ) ) ) ) ).
% frequently_bex_finite
thf(fact_8087_frequently__bex__finite__distrib,axiom,
! [B: $tType,A: $tType,A6: set @ A,P: B > A > $o,F6: filter @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( frequently @ B
@ ^ [X4: B] :
? [Y3: A] :
( ( member @ A @ Y3 @ A6 )
& ( P @ X4 @ Y3 ) )
@ F6 )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( frequently @ B
@ ^ [Y3: B] : ( P @ Y3 @ X4 )
@ F6 ) ) ) ) ) ).
% frequently_bex_finite_distrib
thf(fact_8088_frequently__imp__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F6: filter @ A] :
( ( frequently @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
@ F6 )
= ( ( eventually @ A @ P @ F6 )
=> ( frequently @ A @ Q @ F6 ) ) ) ).
% frequently_imp_iff
thf(fact_8089_frequently__rev__mp,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o] :
( ( frequently @ A @ P @ F6 )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
@ F6 )
=> ( frequently @ A @ Q @ F6 ) ) ) ).
% frequently_rev_mp
thf(fact_8090_not__frequently,axiom,
! [A: $tType,P: A > $o,F6: filter @ A] :
( ( ~ ( frequently @ A @ P @ F6 ) )
= ( eventually @ A
@ ^ [X4: A] :
~ ( P @ X4 )
@ F6 ) ) ).
% not_frequently
thf(fact_8091_not__eventually,axiom,
! [A: $tType,P: A > $o,F6: filter @ A] :
( ( ~ ( eventually @ A @ P @ F6 ) )
= ( frequently @ A
@ ^ [X4: A] :
~ ( P @ X4 )
@ F6 ) ) ).
% not_eventually
thf(fact_8092_frequently__def,axiom,
! [A: $tType] :
( ( frequently @ A )
= ( ^ [P4: A > $o,F9: filter @ A] :
~ ( eventually @ A
@ ^ [X4: A] :
~ ( P4 @ X4 )
@ F9 ) ) ) ).
% frequently_def
thf(fact_8093_frequently__mp,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
@ F6 )
=> ( ( frequently @ A @ P @ F6 )
=> ( frequently @ A @ Q @ F6 ) ) ) ).
% frequently_mp
thf(fact_8094_eventually__frequently__const__simps_I5_J,axiom,
! [A: $tType,P: A > $o,C4: $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> C4 )
@ F6 )
= ( ( frequently @ A @ P @ F6 )
=> C4 ) ) ).
% eventually_frequently_const_simps(5)
thf(fact_8095_not__frequently__False,axiom,
! [A: $tType,F6: filter @ A] :
~ ( frequently @ A
@ ^ [X4: A] : $false
@ F6 ) ).
% not_frequently_False
thf(fact_8096_frequently__disj__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F6: filter @ A] :
( ( frequently @ A
@ ^ [X4: A] :
( ( P @ X4 )
| ( Q @ X4 ) )
@ F6 )
= ( ( frequently @ A @ P @ F6 )
| ( frequently @ A @ Q @ F6 ) ) ) ).
% frequently_disj_iff
thf(fact_8097_frequently__elim1,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o] :
( ( frequently @ A @ P @ F6 )
=> ( ! [I3: A] :
( ( P @ I3 )
=> ( Q @ I3 ) )
=> ( frequently @ A @ Q @ F6 ) ) ) ).
% frequently_elim1
thf(fact_8098_frequently__disj,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o] :
( ( frequently @ A @ P @ F6 )
=> ( ( frequently @ A @ Q @ F6 )
=> ( frequently @ A
@ ^ [X4: A] :
( ( P @ X4 )
| ( Q @ X4 ) )
@ F6 ) ) ) ).
% frequently_disj
thf(fact_8099_frequently__ex,axiom,
! [A: $tType,P: A > $o,F6: filter @ A] :
( ( frequently @ A @ P @ F6 )
=> ? [X_12: A] : ( P @ X_12 ) ) ).
% frequently_ex
thf(fact_8100_eventually__frequently__const__simps_I1_J,axiom,
! [A: $tType,P: A > $o,C4: $o,F6: filter @ A] :
( ( frequently @ A
@ ^ [X4: A] :
( ( P @ X4 )
& C4 )
@ F6 )
= ( ( frequently @ A @ P @ F6 )
& C4 ) ) ).
% eventually_frequently_const_simps(1)
thf(fact_8101_eventually__frequently__const__simps_I2_J,axiom,
! [A: $tType,C4: $o,P: A > $o,F6: filter @ A] :
( ( frequently @ A
@ ^ [X4: A] :
( C4
& ( P @ X4 ) )
@ F6 )
= ( C4
& ( frequently @ A @ P @ F6 ) ) ) ).
% eventually_frequently_const_simps(2)
thf(fact_8102_frequently__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F6: filter @ A] :
( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ( frequently @ A @ P @ F6 )
=> ( frequently @ A @ Q @ F6 ) ) ) ).
% frequently_mono
thf(fact_8103_frequentlyE,axiom,
! [A: $tType,P: A > $o,F6: filter @ A] :
( ( frequently @ A @ P @ F6 )
=> ~ ! [X5: A] :
~ ( P @ X5 ) ) ).
% frequentlyE
thf(fact_8104_frequently__all,axiom,
! [B: $tType,A: $tType,P: A > B > $o,F6: filter @ A] :
( ( frequently @ A
@ ^ [X4: A] :
! [X8: B] : ( P @ X4 @ X8 )
@ F6 )
= ( ! [Y10: A > B] :
( frequently @ A
@ ^ [X4: A] : ( P @ X4 @ ( Y10 @ X4 ) )
@ F6 ) ) ) ).
% frequently_all
thf(fact_8105_eventually__frequentlyE,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F6 )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ~ ( P @ X4 )
| ( Q @ X4 ) )
@ F6 )
=> ( ( F6
!= ( bot_bot @ ( filter @ A ) ) )
=> ( frequently @ A @ Q @ F6 ) ) ) ) ).
% eventually_frequentlyE
thf(fact_8106_eventually__frequently,axiom,
! [A: $tType,F6: filter @ A,P: A > $o] :
( ( F6
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A @ P @ F6 )
=> ( frequently @ A @ P @ F6 ) ) ) ).
% eventually_frequently
thf(fact_8107_frequently__const__iff,axiom,
! [A: $tType,P: $o,F6: filter @ A] :
( ( frequently @ A
@ ^ [X4: A] : P
@ F6 )
= ( P
& ( F6
!= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% frequently_const_iff
thf(fact_8108_le__filter__parametric,axiom,
! [A: $tType,B: $tType,A6: A > B > $o] :
( ( bi_unique @ A @ B @ A6 )
=> ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > $o ) @ ( ( filter @ B ) > $o ) @ ( rel_filter @ A @ B @ A6 )
@ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ $o @ $o @ ( rel_filter @ A @ B @ A6 )
@ ^ [Y5: $o,Z: $o] : Y5 = Z )
@ ( ord_less_eq @ ( filter @ A ) )
@ ( ord_less_eq @ ( filter @ B ) ) ) ) ).
% le_filter_parametric
thf(fact_8109_less__filter__parametric,axiom,
! [A: $tType,B: $tType,A6: A > B > $o] :
( ( bi_unique @ A @ B @ A6 )
=> ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > $o ) @ ( ( filter @ B ) > $o ) @ ( rel_filter @ A @ B @ A6 )
@ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ $o @ $o @ ( rel_filter @ A @ B @ A6 )
@ ^ [Y5: $o,Z: $o] : Y5 = Z )
@ ( ord_less @ ( filter @ A ) )
@ ( ord_less @ ( filter @ B ) ) ) ) ).
% less_filter_parametric
thf(fact_8110_List_Oset__insert,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( set2 @ A @ ( insert @ A @ X3 @ Xs2 ) )
= ( insert2 @ A @ X3 @ ( set2 @ A @ Xs2 ) ) ) ).
% List.set_insert
thf(fact_8111_has__vector__derivative__scaleR,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: real > real,F8: real,X3: real,S: set @ real,G3: real > A,G6: A] :
( ( has_field_derivative @ real @ F3 @ F8 @ ( topolo174197925503356063within @ real @ X3 @ S ) )
=> ( ( has_ve8173657378732805170vative @ A @ G3 @ G6 @ ( topolo174197925503356063within @ real @ X3 @ S ) )
=> ( has_ve8173657378732805170vative @ A
@ ^ [X4: real] : ( real_V8093663219630862766scaleR @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( F3 @ X3 ) @ G6 ) @ ( real_V8093663219630862766scaleR @ A @ F8 @ ( G3 @ X3 ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ S ) ) ) ) ) ).
% has_vector_derivative_scaleR
thf(fact_8112_in__set__insert,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( insert @ A @ X3 @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_8113_not__in__set__insert,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( insert @ A @ X3 @ Xs2 )
= ( cons @ A @ X3 @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_8114_has__vector__derivative__add__const,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: real > A,Z2: A,F8: A,Net: filter @ real] :
( ( has_ve8173657378732805170vative @ A
@ ^ [T3: real] : ( plus_plus @ A @ ( G3 @ T3 ) @ Z2 )
@ F8
@ Net )
= ( has_ve8173657378732805170vative @ A @ G3 @ F8 @ Net ) ) ) ).
% has_vector_derivative_add_const
thf(fact_8115_has__vector__derivative__add,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: real > A,F8: A,Net: filter @ real,G3: real > A,G6: A] :
( ( has_ve8173657378732805170vative @ A @ F3 @ F8 @ Net )
=> ( ( has_ve8173657378732805170vative @ A @ G3 @ G6 @ Net )
=> ( has_ve8173657378732805170vative @ A
@ ^ [X4: real] : ( plus_plus @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( plus_plus @ A @ F8 @ G6 )
@ Net ) ) ) ) ).
% has_vector_derivative_add
thf(fact_8116_List_Oinsert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [X4: A,Xs: list @ A] : ( if @ ( list @ A ) @ ( member @ A @ X4 @ ( set2 @ A @ Xs ) ) @ Xs @ ( cons @ A @ X4 @ Xs ) ) ) ) ).
% List.insert_def
thf(fact_8117_has__vector__derivative__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: real > A,F8: A,X3: real,S: set @ real,G3: real > A,G6: A] :
( ( has_ve8173657378732805170vative @ A @ F3 @ F8 @ ( topolo174197925503356063within @ real @ X3 @ S ) )
=> ( ( has_ve8173657378732805170vative @ A @ G3 @ G6 @ ( topolo174197925503356063within @ real @ X3 @ S ) )
=> ( has_ve8173657378732805170vative @ A
@ ^ [X4: real] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( F3 @ X3 ) @ G6 ) @ ( times_times @ A @ F8 @ ( G3 @ X3 ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ S ) ) ) ) ) ).
% has_vector_derivative_mult
thf(fact_8118_bounded__bilinear_Ohas__vector__derivative,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [Prod: A > B > C,F3: real > A,F8: A,X3: real,S: set @ real,G3: real > B,G6: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( has_ve8173657378732805170vative @ A @ F3 @ F8 @ ( topolo174197925503356063within @ real @ X3 @ S ) )
=> ( ( has_ve8173657378732805170vative @ B @ G3 @ G6 @ ( topolo174197925503356063within @ real @ X3 @ S ) )
=> ( has_ve8173657378732805170vative @ C
@ ^ [X4: real] : ( Prod @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( plus_plus @ C @ ( Prod @ ( F3 @ X3 ) @ G6 ) @ ( Prod @ F8 @ ( G3 @ X3 ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ S ) ) ) ) ) ) ).
% bounded_bilinear.has_vector_derivative
thf(fact_8119_Real_Opositive__def,axiom,
( positive
= ( 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] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ R5 @ ( X8 @ N3 ) ) ) ) ) ) ).
% Real.positive_def
thf(fact_8120_Range__def,axiom,
! [B: $tType,A: $tType] :
( ( range @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ B
@ ( rangep @ A @ B
@ ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R5 ) ) ) ) ) ).
% Range_def
thf(fact_8121_id__funpow,axiom,
! [A: $tType,N: nat] :
( ( compow @ ( A > A ) @ N @ ( id @ A ) )
= ( id @ A ) ) ).
% id_funpow
thf(fact_8122_filtermap__id,axiom,
! [A: $tType] :
( ( filtermap @ A @ A @ ( id @ A ) )
= ( id @ ( filter @ A ) ) ) ).
% filtermap_id
thf(fact_8123_apfst__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apfst @ A @ A @ B @ ( id @ A ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apfst_id
thf(fact_8124_apsnd__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apsnd_id
thf(fact_8125_comp__the__Some,axiom,
! [A: $tType] :
( ( comp @ ( option @ A ) @ A @ A @ ( the2 @ A ) @ ( some @ A ) )
= ( id @ A ) ) ).
% comp_the_Some
thf(fact_8126_snd__diag__id,axiom,
! [A: $tType,Z2: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Z2 )
= ( id @ A @ Z2 ) ) ).
% snd_diag_id
thf(fact_8127_fst__diag__id,axiom,
! [A: $tType,Z2: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Z2 )
= ( id @ A @ Z2 ) ) ).
% fst_diag_id
thf(fact_8128_fold__id,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,F3: A > B > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( ( F3 @ X5 )
= ( id @ B ) ) )
=> ( ( fold @ A @ B @ F3 @ Xs2 )
= ( id @ B ) ) ) ).
% fold_id
thf(fact_8129_apsnd__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apsnd @ B @ C @ A )
= ( product_map_prod @ A @ A @ B @ C @ ( id @ A ) ) ) ).
% apsnd_def
thf(fact_8130_apfst__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apfst @ A @ C @ B )
= ( ^ [F4: A > C] : ( product_map_prod @ A @ C @ B @ B @ F4 @ ( id @ B ) ) ) ) ).
% apfst_def
thf(fact_8131_RangepE,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,B2: B] :
( ( rangep @ A @ B @ R2 @ B2 )
=> ~ ! [A5: A] :
~ ( R2 @ A5 @ B2 ) ) ).
% RangepE
thf(fact_8132_RangePI,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A3: A,B2: B] :
( ( R2 @ A3 @ B2 )
=> ( rangep @ A @ B @ R2 @ B2 ) ) ).
% RangePI
thf(fact_8133_Rangep_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( rangep @ A @ B )
= ( ^ [R5: A > B > $o,A8: B] :
? [B8: A,C6: B] :
( ( A8 = C6 )
& ( R5 @ B8 @ C6 ) ) ) ) ).
% Rangep.simps
thf(fact_8134_Rangep_Ocases,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A3: B] :
( ( rangep @ A @ B @ R2 @ A3 )
=> ~ ! [A5: A] :
~ ( R2 @ A5 @ A3 ) ) ).
% Rangep.cases
thf(fact_8135_option_Omap__id0,axiom,
! [A: $tType] :
( ( map_option @ A @ A @ ( id @ A ) )
= ( id @ ( option @ A ) ) ) ).
% option.map_id0
thf(fact_8136_option_Omap__id,axiom,
! [A: $tType,T2: option @ A] :
( ( map_option @ A @ A @ ( id @ A ) @ T2 )
= T2 ) ).
% option.map_id
thf(fact_8137_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_8138_Rangep__Range__eq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( rangep @ A @ B
@ ^ [X4: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R2 ) )
= ( ^ [X4: B] : ( member @ B @ X4 @ ( range @ A @ B @ R2 ) ) ) ) ).
% Rangep_Range_eq
thf(fact_8139_ofilter__embed,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A6 )
= ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( field2 @ A @ R2 ) )
& ( bNF_Wellorder_embed @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu3: A] : A6 ) )
@ R2
@ ( id @ A ) ) ) ) ) ).
% ofilter_embed
thf(fact_8140_ofilter__subset__embed,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A6: set @ A,B5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A6 )
=> ( ( order_ofilter @ A @ R2 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
= ( bNF_Wellorder_embed @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu3: A] : A6 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B5
@ ^ [Uu3: A] : B5 ) )
@ ( id @ A ) ) ) ) ) ) ).
% ofilter_subset_embed
thf(fact_8141_of__nat__eq__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( id @ nat ) ) ).
% of_nat_eq_id
thf(fact_8142_case__prod__Pair,axiom,
! [B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% case_prod_Pair
thf(fact_8143_swap__comp__swap,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( product_swap @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% swap_comp_swap
thf(fact_8144_map__prod_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X4: A] : X4
@ ^ [X4: B] : X4 )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% map_prod.identity
thf(fact_8145_map__option_Oidentity,axiom,
! [A: $tType] :
( ( map_option @ A @ A
@ ^ [X4: A] : X4 )
= ( id @ ( option @ A ) ) ) ).
% map_option.identity
thf(fact_8146_ordLess__not__embed,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R4: 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 ) ) @ R2 @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ~ ? [X_1: B > A] : ( bNF_Wellorder_embed @ B @ A @ R4 @ R2 @ X_1 ) ) ).
% ordLess_not_embed
thf(fact_8147_BNF__Wellorder__Constructions_OordLess__Field,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F3: 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 ) ) @ R1 @ R22 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( bNF_Wellorder_embed @ A @ B @ R1 @ R22 @ F3 )
=> ( ( image2 @ A @ B @ F3 @ ( field2 @ A @ R1 ) )
!= ( field2 @ B @ R22 ) ) ) ) ).
% BNF_Wellorder_Constructions.ordLess_Field
thf(fact_8148_ordLess__iff,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R4: 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 ) ) @ R2 @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
= ( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
& ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
& ~ ? [X8: B > A] : ( bNF_Wellorder_embed @ B @ A @ R4 @ R2 @ X8 ) ) ) ).
% ordLess_iff
thf(fact_8149_embed__ordLess__ofilterIncl,axiom,
! [B: $tType,A: $tType,C: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R32: set @ ( product_prod @ C @ C ),F132: A > C,F232: B > 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 ) ) @ R1 @ R22 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R22 @ R32 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( ( bNF_Wellorder_embed @ A @ C @ R1 @ R32 @ F132 )
=> ( ( bNF_Wellorder_embed @ B @ C @ R22 @ R32 @ F232 )
=> ( member @ ( product_prod @ ( set @ C ) @ ( set @ C ) ) @ ( product_Pair @ ( set @ C ) @ ( set @ C ) @ ( image2 @ A @ C @ F132 @ ( field2 @ A @ R1 ) ) @ ( image2 @ B @ C @ F232 @ ( field2 @ B @ R22 ) ) ) @ ( bNF_We413866401316099525erIncl @ C @ R32 ) ) ) ) ) ) ).
% embed_ordLess_ofilterIncl
thf(fact_8150_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_8151_aboveS__def,axiom,
! [A: $tType] :
( ( order_aboveS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A8: A] :
( collect @ A
@ ^ [B8: A] :
( ( B8 != A8 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B8 ) @ R5 ) ) ) ) ) ).
% aboveS_def
thf(fact_8152_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A7: set @ A] :
( A7
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_8153_is__empty__set,axiom,
! [A: $tType,Xs2: list @ A] :
( ( is_empty @ A @ ( set2 @ A @ Xs2 ) )
= ( null @ A @ Xs2 ) ) ).
% is_empty_set
thf(fact_8154_relInvImage__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr7122648621184425601vImage @ A @ B )
= ( ^ [A7: set @ A,R6: set @ ( product_prod @ B @ B ),F4: A > B] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [A12: A,A23: A] :
( ( Uu3
= ( product_Pair @ A @ A @ A12 @ A23 ) )
& ( member @ A @ A12 @ A7 )
& ( member @ A @ A23 @ A7 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ A12 ) @ ( F4 @ A23 ) ) @ R6 ) ) ) ) ) ).
% relInvImage_def
thf(fact_8155_scomp__unfold,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( G4 @ ( product_fst @ B @ C @ ( F4 @ X4 ) ) @ ( product_snd @ B @ C @ ( F4 @ X4 ) ) ) ) ) ).
% scomp_unfold
thf(fact_8156_semilattice__order__set_Osubset__imp,axiom,
! [A: $tType,F3: A > A > A,Less_eq2: A > A > $o,Less: A > A > $o,A6: set @ A,B5: set @ A] :
( ( lattic4895041142388067077er_set @ A @ F3 @ Less_eq2 @ Less )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( Less_eq2 @ ( lattic1715443433743089157tice_F @ A @ F3 @ B5 ) @ ( lattic1715443433743089157tice_F @ A @ F3 @ A6 ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_8157_scomp__apply,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_scomp @ B @ C @ D @ A )
= ( ^ [F4: B > ( product_prod @ C @ D ),G4: C > D > A,X4: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F4 @ X4 ) ) ) ) ).
% scomp_apply
thf(fact_8158_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: $tType,E: $tType,F3: A > ( product_prod @ E @ F ),G3: E > F > ( product_prod @ C @ D ),H: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F @ ( product_prod @ C @ D ) @ F3 @ G3 ) @ H )
= ( product_scomp @ A @ E @ F @ B @ F3
@ ^ [X4: E] : ( product_scomp @ F @ C @ D @ B @ ( G3 @ X4 ) @ H ) ) ) ).
% scomp_scomp
thf(fact_8159_scomp__def,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F4 @ X4 ) ) ) ) ).
% scomp_def
thf(fact_8160_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X3: A > ( product_prod @ B @ C )] :
( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X3 @ ( product_Pair @ B @ C ) )
= X3 ) ).
% scomp_Pair
thf(fact_8161_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X3: C,F3: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X3 ) @ F3 )
= ( F3 @ X3 ) ) ).
% Pair_scomp
thf(fact_8162_semilattice__order__set_OboundedE,axiom,
! [A: $tType,F3: A > A > A,Less_eq2: A > A > $o,Less: A > A > $o,A6: set @ A,X3: A] :
( ( lattic4895041142388067077er_set @ A @ F3 @ Less_eq2 @ Less )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq2 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ A6 ) )
=> ! [A18: A] :
( ( member @ A @ A18 @ A6 )
=> ( Less_eq2 @ X3 @ A18 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_8163_semilattice__order__set_OboundedI,axiom,
! [A: $tType,F3: A > A > A,Less_eq2: A > A > $o,Less: A > A > $o,A6: set @ A,X3: A] :
( ( lattic4895041142388067077er_set @ A @ F3 @ Less_eq2 @ Less )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ( Less_eq2 @ X3 @ A5 ) )
=> ( Less_eq2 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ A6 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_8164_semilattice__order__set_Obounded__iff,axiom,
! [A: $tType,F3: A > A > A,Less_eq2: A > A > $o,Less: A > A > $o,A6: set @ A,X3: A] :
( ( lattic4895041142388067077er_set @ A @ F3 @ Less_eq2 @ Less )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq2 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ A6 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( Less_eq2 @ X3 @ X4 ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_8165_semilattice__set_Oeq__fold_H,axiom,
! [A: $tType,F3: A > A > A,A6: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ A6 )
= ( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( F3 @ X4 ) @ Y3 ) )
@ ( none @ A )
@ A6 ) ) ) ) ).
% semilattice_set.eq_fold'
thf(fact_8166_semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F3: A > A > A,A6: set @ A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert2 @ A @ X3 @ A6 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert2 @ A @ X3 @ A6 ) )
= ( F3 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_8167_semilattice__set_Osingleton,axiom,
! [A: $tType,F3: A > A > A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% semilattice_set.singleton
thf(fact_8168_semilattice__set_Ohom__commute,axiom,
! [A: $tType,F3: A > A > A,H: A > A,N5: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ! [X5: A,Y4: A] :
( ( H @ ( F3 @ X5 @ Y4 ) )
= ( F3 @ ( H @ X5 ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite2 @ A @ N5 )
=> ( ( N5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic1715443433743089157tice_F @ A @ F3 @ N5 ) )
= ( lattic1715443433743089157tice_F @ A @ F3 @ ( image2 @ A @ A @ H @ N5 ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_8169_semilattice__set_Osubset,axiom,
! [A: $tType,F3: A > A > A,A6: set @ A,B5: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ( F3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ B5 ) @ ( lattic1715443433743089157tice_F @ A @ F3 @ A6 ) )
= ( lattic1715443433743089157tice_F @ A @ F3 @ A6 ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_8170_semilattice__set_Oinsert__not__elem,axiom,
! [A: $tType,F3: A > A > A,A6: set @ A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X3 @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert2 @ A @ X3 @ A6 ) )
= ( F3 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ A6 ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_8171_semilattice__set_Oinsert,axiom,
! [A: $tType,F3: A > A > A,A6: set @ A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert2 @ A @ X3 @ A6 ) )
= ( F3 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ A6 ) ) ) ) ) ) ).
% semilattice_set.insert
thf(fact_8172_semilattice__set_Oclosed,axiom,
! [A: $tType,F3: A > A > A,A6: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y4: A] : ( member @ A @ ( F3 @ X5 @ Y4 ) @ ( insert2 @ A @ X5 @ ( insert2 @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic1715443433743089157tice_F @ A @ F3 @ A6 ) @ A6 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_8173_semilattice__set_Ounion,axiom,
! [A: $tType,F3: A > A > A,A6: set @ A,B5: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( sup_sup @ ( set @ A ) @ A6 @ B5 ) )
= ( F3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ A6 ) @ ( lattic1715443433743089157tice_F @ A @ F3 @ B5 ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_8174_semilattice__set_Oinfinite,axiom,
! [A: $tType,F3: A > A > A,A6: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ~ ( finite_finite2 @ A @ A6 )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ A6 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% semilattice_set.infinite
thf(fact_8175_semilattice__set_Oeq__fold,axiom,
! [A: $tType,F3: A > A > A,A6: set @ A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert2 @ A @ X3 @ A6 ) )
= ( finite_fold @ A @ A @ F3 @ X3 @ A6 ) ) ) ) ).
% semilattice_set.eq_fold
thf(fact_8176_semilattice__set_Oset__eq__fold,axiom,
! [A: $tType,F3: A > A > A,X3: A,Xs2: list @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( set2 @ A @ ( cons @ A @ X3 @ Xs2 ) ) )
= ( fold @ A @ A @ F3 @ Xs2 @ X3 ) ) ) ).
% semilattice_set.set_eq_fold
thf(fact_8177_semilattice__set_Oremove,axiom,
! [A: $tType,F3: A > A > A,A6: set @ A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ A6 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ A6 )
= ( F3 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_8178_normalize__stable,axiom,
! [Q3: int,P2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Q3 )
=> ( ( algebr8660921524188924756oprime @ int @ P2 @ Q3 )
=> ( ( normalize @ ( product_Pair @ int @ int @ P2 @ Q3 ) )
= ( product_Pair @ int @ int @ P2 @ Q3 ) ) ) ) ).
% normalize_stable
thf(fact_8179_antisymp__antisym__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( antisymp @ A
@ ^ [X4: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 ) )
= ( antisym @ A @ R2 ) ) ).
% antisymp_antisym_eq
thf(fact_8180_coprime__power__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( power_power @ A @ B2 @ N ) )
= ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% coprime_power_right_iff
thf(fact_8181_coprime__power__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( power_power @ A @ A3 @ N ) @ B2 )
= ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% coprime_power_left_iff
thf(fact_8182_coprime__right__2__iff__odd,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% coprime_right_2_iff_odd
% Type constructors (780)
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,axiom,
! [A17: $tType,A26: $tType] :
( ( comple592849572758109894attice @ A26 )
=> ( counta4013691401010221786attice @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A17: $tType,A26: $tType] :
( ( comple6319245703460814977attice @ A26 )
=> ( condit1219197933456340205attice @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
! [A17: $tType,A26: $tType] :
( ( counta3822494911875563373attice @ A26 )
=> ( counta3822494911875563373attice @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A17: $tType,A26: $tType] :
( ( comple592849572758109894attice @ A26 )
=> ( comple592849572758109894attice @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A17: $tType,A26: $tType] :
( ( bounded_lattice @ A26 )
=> ( bounde4967611905675639751up_bot @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A17: $tType,A26: $tType] :
( ( bounded_lattice @ A26 )
=> ( bounde4346867609351753570nf_top @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A17: $tType,A26: $tType] :
( ( comple6319245703460814977attice @ A26 )
=> ( comple6319245703460814977attice @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A17: $tType,A26: $tType] :
( ( boolea8198339166811842893lgebra @ A26 )
=> ( boolea8198339166811842893lgebra @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A17: $tType,A26: $tType] :
( ( bounded_lattice @ A26 )
=> ( bounded_lattice_bot @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A17: $tType,A26: $tType] :
( ( comple6319245703460814977attice @ A26 )
=> ( comple9053668089753744459l_ccpo @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A17: $tType,A26: $tType] :
( ( semilattice_sup @ A26 )
=> ( semilattice_sup @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A17: $tType,A26: $tType] :
( ( semilattice_inf @ A26 )
=> ( semilattice_inf @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A17: $tType,A26: $tType] :
( ( distrib_lattice @ A26 )
=> ( distrib_lattice @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A17: $tType,A26: $tType] :
( ( bounded_lattice @ A26 )
=> ( bounded_lattice @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A17: $tType,A26: $tType] :
( ( order_top @ A26 )
=> ( order_top @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A17: $tType,A26: $tType] :
( ( order_bot @ A26 )
=> ( order_bot @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A17: $tType,A26: $tType] :
( ( preorder @ A26 )
=> ( preorder @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A17: $tType,A26: $tType] :
( ( ( finite_finite @ A17 )
& ( finite_finite @ A26 ) )
=> ( finite_finite @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A17: $tType,A26: $tType] :
( ( lattice @ A26 )
=> ( lattice @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A17: $tType,A26: $tType] :
( ( order @ A26 )
=> ( order @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A17: $tType,A26: $tType] :
( ( top @ A26 )
=> ( top @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A17: $tType,A26: $tType] :
( ( ord @ A26 )
=> ( ord @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A17: $tType,A26: $tType] :
( ( bot @ A26 )
=> ( bot @ ( A17 > A26 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A17: $tType,A26: $tType] :
( ( uminus @ A26 )
=> ( uminus @ ( A17 > A26 ) ) ) ).
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___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
euclid8789492081693882211th_nat @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
euclid3128863361964157862miring @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___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___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___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri3467727345109120633visors @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere6658533253407199908up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere166539214618696060dd_abs @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
semiri6843258321239162965malize @ int ).
thf(tcon_Int_Oint___Limits_Otopological__monoid__mult,axiom,
topolo1898628316856586783d_mult @ int ).
thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel2418104881723323429up_add @ int ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors @ int ).
thf(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
topolo6943815403480290642id_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1802427076303600483id_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord4710134922213307826strict @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
comm_s4317794764714335236cancel @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
bit_semiring_bits @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
topological_t2_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Ot1__space,axiom,
topological_t1_space @ int ).
thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere2520102378445227354miring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_2,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_3,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Lattices_Odistrib__lattice_4,axiom,
distrib_lattice @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
semiring_1_cancel @ int ).
thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
algebraic_semidom @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0 @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
semidom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int ).
thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring @ int ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring @ int ).
thf(tcon_Int_Oint___Parity_Oring__parity,axiom,
ring_parity @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_5,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0 @ int ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int ).
thf(tcon_Int_Oint___Lattices_Olattice_6,axiom,
lattice @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
semiring_gcd @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
semiring_Gcd @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
comm_ring @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_7,axiom,
order @ int ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring @ int ).
thf(tcon_Int_Oint___Orderings_Oord_8,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_9,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___GCD_Oring__gcd,axiom,
ring_gcd @ int ).
thf(tcon_Int_Oint___Power_Opower,axiom,
power @ int ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Groups_Oplus,axiom,
plus @ int ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom,axiom,
idom @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Int_Oint___Rings_Odvd,axiom,
dvd @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_10,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_12,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_13,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_14,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_15,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_16,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_17,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_18,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_19,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_20,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_21,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_22,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_23,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_24,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_25,axiom,
ordere8940638589300402666id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni5634975068530333245id_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_26,axiom,
topolo4958980785337419405_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_27,axiom,
topolo1944317154257567458pology @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_28,axiom,
topolo8865339358273720382pology @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_29,axiom,
topolo5987344860129210374id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_30,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_31,axiom,
topolo2564578578187576103pology @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_32,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_33,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_34,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_35,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_36,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_37,axiom,
semiri6843258321239162965malize @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__mult_38,axiom,
topolo1898628316856586783d_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_39,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_40,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_41,axiom,
topolo6943815403480290642id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_42,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_43,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_44,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_45,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot1__space_46,axiom,
topological_t1_space @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_47,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_48,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_49,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_50,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_51,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_52,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_53,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_54,axiom,
distrib_lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_55,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_56,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_57,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_58,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_59,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_60,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_61,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_62,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_63,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_64,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_65,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_66,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_67,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_68,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_69,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_70,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_71,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_72,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_73,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_74,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_75,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_76,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_77,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_78,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_79,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_80,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_81,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_82,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_83,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_84,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_85,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_86,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_87,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_88,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Power_Opower_89,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_90,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_91,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_92,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_93,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_94,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_95,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_96,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_97,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_98,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Oplus_99,axiom,
plus @ num ).
thf(tcon_Num_Onum___Nat_Osize_100,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_101,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_102,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_103,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_104,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_105,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_106,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_107,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_108,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_109,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_110,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_111,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_112,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_113,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_114,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_115,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_116,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_117,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_118,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_119,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_120,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_121,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_122,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_123,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_124,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_125,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_126,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_127,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_128,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_129,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_130,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_131,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_132,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_133,axiom,
distrib_lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_134,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_135,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_136,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_137,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_138,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_139,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_140,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_141,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_142,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_143,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_144,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_145,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_146,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_147,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_148,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_149,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_150,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_151,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_152,axiom,
ab_group_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_153,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_154,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_155,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_156,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_157,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_158,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_159,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_160,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_161,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_162,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_163,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_164,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_165,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_166,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_167,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_168,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_169,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_170,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_171,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_172,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_173,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_174,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_175,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_176,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_177,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_178,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_179,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_180,axiom,
dvd @ rat ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_181,axiom,
! [A17: $tType] : ( counta4013691401010221786attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_182,axiom,
! [A17: $tType] : ( condit1219197933456340205attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_183,axiom,
! [A17: $tType] : ( counta3822494911875563373attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_184,axiom,
! [A17: $tType] : ( comple592849572758109894attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_185,axiom,
! [A17: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_186,axiom,
! [A17: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_187,axiom,
! [A17: $tType] : ( comple6319245703460814977attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_188,axiom,
! [A17: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_189,axiom,
! [A17: $tType] : ( bounded_lattice_bot @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_190,axiom,
! [A17: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_191,axiom,
! [A17: $tType] : ( semilattice_sup @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_192,axiom,
! [A17: $tType] : ( semilattice_inf @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_193,axiom,
! [A17: $tType] : ( distrib_lattice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_194,axiom,
! [A17: $tType] : ( bounded_lattice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_195,axiom,
! [A17: $tType] : ( order_top @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_196,axiom,
! [A17: $tType] : ( order_bot @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_197,axiom,
! [A17: $tType] : ( preorder @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_198,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( finite_finite @ ( set @ A17 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_199,axiom,
! [A17: $tType] : ( lattice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_200,axiom,
! [A17: $tType] : ( order @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_201,axiom,
! [A17: $tType] : ( top @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_202,axiom,
! [A17: $tType] : ( ord @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_203,axiom,
! [A17: $tType] : ( bot @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_204,axiom,
! [A17: $tType] : ( uminus @ ( set @ A17 ) ) ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_205,axiom,
counta4013691401010221786attice @ $o ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_206,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_207,axiom,
counta3822494911875563373attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_208,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_209,axiom,
topolo4958980785337419405_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_210,axiom,
topolo1944317154257567458pology @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_211,axiom,
topolo8865339358273720382pology @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_212,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_213,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_214,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_215,axiom,
topolo2564578578187576103pology @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_216,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_217,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_218,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_219,axiom,
topological_t1_space @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_220,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_221,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_222,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_223,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_224,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_225,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_226,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_227,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_228,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_229,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_230,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_231,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_232,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_233,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_234,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_235,axiom,
uminus @ $o ).
thf(tcon_List_Olist___Nat_Osize_236,axiom,
! [A17: $tType] : ( size @ ( list @ A17 ) ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_237,axiom,
condit6923001295902523014norder @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_238,axiom,
condit1219197933456340205attice @ real ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_239,axiom,
semiri1453513574482234551roduct @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_240,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___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_241,axiom,
strict9044650504122735259up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_242,axiom,
ordere580206878836729694up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_243,axiom,
ordere2412721322843649153imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_244,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_245,axiom,
strict7427464778891057005id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_246,axiom,
ordere8940638589300402666id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_247,axiom,
topolo4958980785337419405_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_248,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_249,axiom,
archim462609752435547400_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_250,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_251,axiom,
unboun7993243217541854897norder @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_252,axiom,
topolo5987344860129210374id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_253,axiom,
linord4140545234300271783up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_254,axiom,
topolo2564578578187576103pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_255,axiom,
semiri2026040879449505780visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_256,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___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_257,axiom,
linord8928482502909563296strict @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_258,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_259,axiom,
ordere6658533253407199908up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_260,axiom,
ordere166539214618696060dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_261,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_262,axiom,
ordere6911136660526730532id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_263,axiom,
linord5086331880401160121up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_264,axiom,
cancel2418104881723323429up_add @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_265,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_266,axiom,
topolo6943815403480290642id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_267,axiom,
cancel1802427076303600483id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_268,axiom,
linord4710134922213307826strict @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_269,axiom,
comm_s4317794764714335236cancel @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
topolo1633459387980952147up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_270,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot1__space_271,axiom,
topological_t1_space @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_272,axiom,
ordere2520102378445227354miring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_273,axiom,
linord6961819062388156250ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_274,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_275,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring_276,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_277,axiom,
ordered_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_278,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder_279,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_280,axiom,
semilattice_sup @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__inf_281,axiom,
semilattice_inf @ real ).
thf(tcon_Real_Oreal___Lattices_Odistrib__lattice_282,axiom,
distrib_lattice @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_283,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_284,axiom,
semiring_1_cancel @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_285,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_286,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field_287,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_288,axiom,
ordered_semiring @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_289,axiom,
ordered_ring_abs @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_290,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_291,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_292,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_293,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_294,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order_295,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_296,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_297,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_298,axiom,
semiring_numeral @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_299,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring_300,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_301,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring_302,axiom,
comm_semiring @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_303,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_304,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0_305,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_306,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_307,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_308,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_309,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_310,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_311,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_312,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_313,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_314,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_315,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_316,axiom,
lattice @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_317,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring_318,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_319,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_320,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_321,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring_322,axiom,
semiring @ real ).
thf(tcon_Real_Oreal___Fields_Oinverse_323,axiom,
inverse @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_324,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_325,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_326,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Fields_Ofield_327,axiom,
field @ real ).
thf(tcon_Real_Oreal___Power_Opower_328,axiom,
power @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_329,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_330,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_331,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Rings_Oring_332,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom_333,axiom,
idom @ real ).
thf(tcon_Real_Oreal___Groups_Oone_334,axiom,
one @ real ).
thf(tcon_Real_Oreal___Rings_Odvd_335,axiom,
dvd @ real ).
thf(tcon_String_Ochar___Finite__Set_Ofinite_336,axiom,
finite_finite @ char ).
thf(tcon_String_Ochar___Nat_Osize_337,axiom,
size @ char ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_338,axiom,
! [A17: $tType,A26: $tType] :
( ( ( finite_finite @ A17 )
& ( finite_finite @ A26 ) )
=> ( finite_finite @ ( sum_sum @ A17 @ A26 ) ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_339,axiom,
! [A17: $tType,A26: $tType] : ( size @ ( sum_sum @ A17 @ A26 ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_340,axiom,
! [A17: $tType] : ( condit1219197933456340205attice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_341,axiom,
! [A17: $tType] : ( counta3822494911875563373attice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_342,axiom,
! [A17: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_343,axiom,
! [A17: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_344,axiom,
! [A17: $tType] : ( comple6319245703460814977attice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_345,axiom,
! [A17: $tType] : ( bounded_lattice_bot @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_346,axiom,
! [A17: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_347,axiom,
! [A17: $tType] : ( semilattice_sup @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_348,axiom,
! [A17: $tType] : ( semilattice_inf @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_349,axiom,
! [A17: $tType] : ( distrib_lattice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_350,axiom,
! [A17: $tType] : ( bounded_lattice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_351,axiom,
! [A17: $tType] : ( order_top @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_352,axiom,
! [A17: $tType] : ( order_bot @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_353,axiom,
! [A17: $tType] : ( preorder @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_354,axiom,
! [A17: $tType] : ( lattice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_355,axiom,
! [A17: $tType] : ( order @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_356,axiom,
! [A17: $tType] : ( top @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_357,axiom,
! [A17: $tType] : ( ord @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_358,axiom,
! [A17: $tType] : ( bot @ ( filter @ A17 ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_359,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( finite_finite @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_360,axiom,
! [A17: $tType] : ( size @ ( option @ A17 ) ) ).
thf(tcon_String_Oliteral___Groups_Osemigroup__add_361,axiom,
semigroup_add @ literal ).
thf(tcon_String_Oliteral___Orderings_Opreorder_362,axiom,
preorder @ literal ).
thf(tcon_String_Oliteral___Orderings_Olinorder_363,axiom,
linorder @ literal ).
thf(tcon_String_Oliteral___Groups_Omonoid__add_364,axiom,
monoid_add @ literal ).
thf(tcon_String_Oliteral___Orderings_Oorder_365,axiom,
order @ literal ).
thf(tcon_String_Oliteral___Orderings_Oord_366,axiom,
ord @ literal ).
thf(tcon_String_Oliteral___Groups_Ozero_367,axiom,
zero @ literal ).
thf(tcon_String_Oliteral___Groups_Oplus_368,axiom,
plus @ literal ).
thf(tcon_String_Oliteral___Nat_Osize_369,axiom,
size @ literal ).
thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_370,axiom,
semiri1453513574482234551roduct @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_371,axiom,
topolo3112930676232923870pology @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_372,axiom,
real_V8999393235501362500lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_373,axiom,
real_V2822296259951069270ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_374,axiom,
real_V4412858255891104859lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_375,axiom,
real_V822414075346904944vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_376,axiom,
topolo4958980785337419405_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_377,axiom,
real_V3459762299906320749_field @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_378,axiom,
real_V5047593784448816457lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_379,axiom,
real_V768167426530841204y_dist @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_380,axiom,
topolo5987344860129210374id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_381,axiom,
semiri2026040879449505780visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_382,axiom,
real_V2191834092415804123ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_383,axiom,
real_V8037385150606011577_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_384,axiom,
topolo7287701948861334536_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_385,axiom,
topolo8386298272705272623_space @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_386,axiom,
semiri3467727345109120633visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_387,axiom,
real_V7819770556892013058_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_388,axiom,
topolo1287966508704411220up_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_389,axiom,
real_V4867850818363320053vector @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_390,axiom,
cancel2418104881723323429up_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_391,axiom,
ring_15535105094025558882visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_392,axiom,
real_V7773925162809079976_field @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_393,axiom,
topolo6943815403480290642id_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_394,axiom,
cancel1802427076303600483id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_395,axiom,
comm_s4317794764714335236cancel @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__group__add_396,axiom,
topolo1633459387980952147up_add @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_397,axiom,
topological_t2_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_398,axiom,
topological_t1_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_399,axiom,
cancel_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_400,axiom,
real_Vector_banach @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_401,axiom,
ab_semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_402,axiom,
semiring_1_cancel @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_403,axiom,
comm_monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_404,axiom,
ab_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_405,axiom,
comm_monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_406,axiom,
comm_semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_407,axiom,
comm_semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_408,axiom,
semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_409,axiom,
semidom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_410,axiom,
semiring_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_411,axiom,
semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_412,axiom,
division_ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_413,axiom,
comm_semiring @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_414,axiom,
semiring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_415,axiom,
ab_group_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_416,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_417,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_418,axiom,
comm_ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_419,axiom,
monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_420,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_421,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_422,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_423,axiom,
comm_ring @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_424,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_425,axiom,
ring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring_426,axiom,
semiring @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Oinverse_427,axiom,
inverse @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ouminus_428,axiom,
uminus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_429,axiom,
ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_430,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Power_Opower_431,axiom,
power @ complex ).
thf(tcon_Complex_Ocomplex___Num_Onumeral_432,axiom,
numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_433,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oplus_434,axiom,
plus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring_435,axiom,
ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom_436,axiom,
idom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_437,axiom,
one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Odvd_438,axiom,
dvd @ complex ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_439,axiom,
condit6923001295902523014norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_440,axiom,
counta4013691401010221786attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_441,axiom,
condit1219197933456340205attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_442,axiom,
counta3822494911875563373attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_443,axiom,
comple592849572758109894attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_444,axiom,
strict9044650504122735259up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_445,axiom,
strict7427464778891057005id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_446,axiom,
canoni5634975068530333245id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_447,axiom,
bounde4967611905675639751up_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_448,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_449,axiom,
linord4140545234300271783up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_450,axiom,
comple6319245703460814977attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_451,axiom,
linord181362715937106298miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_452,axiom,
semiri3467727345109120633visors @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_453,axiom,
ordere6658533253407199908up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_454,axiom,
ordere6911136660526730532id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_455,axiom,
bounded_lattice_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_456,axiom,
ordere2520102378445227354miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_457,axiom,
comple9053668089753744459l_ccpo @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_458,axiom,
semilattice_sup @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_459,axiom,
semilattice_inf @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_460,axiom,
distrib_lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_461,axiom,
bounded_lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_462,axiom,
ab_semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_463,axiom,
comm_monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_464,axiom,
ab_semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_465,axiom,
ordered_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_466,axiom,
comm_monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_467,axiom,
comm_semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_468,axiom,
comm_semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_469,axiom,
semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_470,axiom,
semiring_numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_471,axiom,
semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_472,axiom,
zero_less_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_473,axiom,
comm_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_474,axiom,
wellorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_475,axiom,
order_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_476,axiom,
order_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_477,axiom,
semiring_char_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_478,axiom,
preorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_479,axiom,
linorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_480,axiom,
monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_481,axiom,
monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_482,axiom,
semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_483,axiom,
semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_484,axiom,
lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_485,axiom,
order @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_486,axiom,
semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Otop_487,axiom,
top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_488,axiom,
ord @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Obot_489,axiom,
bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Power_Opower_490,axiom,
power @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_491,axiom,
numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_492,axiom,
zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oplus_493,axiom,
plus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oone_494,axiom,
one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Odvd_495,axiom,
dvd @ extended_enat ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_496,axiom,
! [A17: $tType,A26: $tType] :
( ( ( topolo4958980785337419405_space @ A17 )
& ( topolo4958980785337419405_space @ A26 ) )
=> ( topolo4958980785337419405_space @ ( product_prod @ A17 @ A26 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_497,axiom,
! [A17: $tType,A26: $tType] :
( ( ( topological_t2_space @ A17 )
& ( topological_t2_space @ A26 ) )
=> ( topological_t2_space @ ( product_prod @ A17 @ A26 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_498,axiom,
! [A17: $tType,A26: $tType] :
( ( ( topological_t1_space @ A17 )
& ( topological_t1_space @ A26 ) )
=> ( topological_t1_space @ ( product_prod @ A17 @ A26 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_499,axiom,
! [A17: $tType,A26: $tType] :
( ( ( finite_finite @ A17 )
& ( finite_finite @ A26 ) )
=> ( finite_finite @ ( product_prod @ A17 @ A26 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_500,axiom,
! [A17: $tType,A26: $tType] : ( size @ ( product_prod @ A17 @ A26 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_501,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_502,axiom,
counta4013691401010221786attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_503,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_504,axiom,
counta3822494911875563373attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_505,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_506,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_507,axiom,
bounde4346867609351753570nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_508,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_509,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_510,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_511,axiom,
bounded_lattice_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_512,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_513,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_514,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_515,axiom,
distrib_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_516,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_517,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_518,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_519,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_520,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_521,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_522,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_523,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_524,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_525,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_526,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_527,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_528,axiom,
uminus @ product_unit ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_529,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_530,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_531,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_532,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_533,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_534,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_535,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_536,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_537,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_538,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_539,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_540,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_541,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_542,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_543,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_544,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_545,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_546,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_547,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_548,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_549,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_550,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_551,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_552,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_553,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_554,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_555,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_556,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_557,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_558,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_559,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_560,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_561,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_562,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_563,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_564,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_565,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_566,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_567,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_568,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_569,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_570,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_571,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_572,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_573,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_574,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_575,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_576,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_577,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_578,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_579,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_580,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_581,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_582,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_583,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_584,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_585,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_586,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_587,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_588,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_589,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_590,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_591,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_592,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_593,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_594,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_595,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_596,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_597,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_598,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_599,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_600,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_601,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_602,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_603,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_604,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_605,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_606,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_607,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_608,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_609,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_610,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_611,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_612,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_613,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_614,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_615,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_616,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_617,axiom,
dvd @ code_integer ).
thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_618,axiom,
size @ vEBT_VEBT ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( sa
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ deg @ treeList2 @ summary2 ) ) ).
%------------------------------------------------------------------------------